Intrinsic structure of the cotangent bundle of a locally Euclidean differential space

Abstract

In this paper, the intrinsic structure of the cotangent bundle $T^{*}S$ of a locally Euclidean differential space S is studied. If the canonical 2-form ${ɷ}_S$ of $T^{*}S$ is non-degenerate, then $(T^{*}S,{ɷ}_S)$ is a symplectic differential space. It has a partition by presymplectic manifolds $T_M^{*}S$ , where M is a submanifold of S, $T_M^{*}S={\pi }_S^{-1}(M)$ and ${\pi }_S:T^{*}S\to S$ is the cotangent bundle projection. The corresponding reduced symplectic spaces are symplectic manifolds $(T^{*}M,{ɷ}_M)$ .

1. Introduction

Modern differential geometry is usually restricted to the study of geometric structure of manifolds. The theory of differential spaces deals with applications of the techniques of differential geometry to a large class of singular spaces. It consists of endowing a topological space S with a differential structure $C^{\infty }(S)$ , consisting of functions deemed to be smooth and the study of geometric concepts on S that can be defined in terms of $C^{\infty }(S)$ . The theory of differential spaces was introduced in 1967 by Sikorski [1]. A comprehensive presentation of Sikorski’s theory is contained in his book ‘Introduction to Differential Geometry,’ published in Polish in 1972 [2], in which he studies smooth manifolds as special cases of differential spaces.

There were several mathematicians who preceeded Sikorski in the study of singular spaces. The first theory of manifolds with singularities is the work of Satake, who introduced in 1955 the notion of a V-manifold, defined in terms of an atlas of charts with values in quotients of connected open subsets of ${\mathrm{ℝ}}^n$ by a finite group of linear transformations [3]. In 1961, Cerf introduced the notion-generalized manifold, now known as manifold with corners, defined in terms of an atlas of charts with values in open subsets of $[0,\infty {)}^k\times {\mathrm{ℝ}}^{n-k}\subseteq {\mathrm{ℝ}}^n$ , where $k=0,1,\dots ,n$ [4]. In 1966, Smith introduced his notion of differentiable structure on a topological space, which consists of a family of continuous functions, deemed to be smooth, which carry all the information about the geometry of the space [5]. Smith used the term differentiable spaces, and he studied the de Rham theorem on differentiable spaces.

There are several more differential geometric approaches to singular spaces, e.g., Aronszajn’s subcartesian spaces [6], Spallek’s differentiable spaces [7], and Souriau’s diffeologies [8]. Watts [9] proved that a subcartesian space of Aronszajn is equivalent to a Hausdorff, locally Euclidean differential space. In the literature on differential spaces, the term ‘subcartesian space’ is used to mean a Hausdorff, locally Euclidean differential space. However, in this paper, we adopt some definitions different that are made by in the theory of subcartesian spaces as presented by Marshall [10]. ¹ Therefore, we use here the term ‘Hausdorff locally Euclidean differential space’ rather than ‘subcartesian space’ in order to avoid implications that our results apply to the study of subcartesian spaces.

In this paper, we use the results of Bates et al. [11] and of Cushman and Śniatycki [12] to get an understanding of the structure of the cotangent bundle $T^{*}S$ of a locally Euclidean differential space S. Cushman and Śniatycki [12] discussed the partition $\mathfrak{M}(S)$ of locally Euclidean differential space S by the orbits of the Lie algebra $\mathfrak{X}(S)$ of all vector fields on S, which is a generalization of the notion of Whitney stratification. We apply this result here to study intrinsic differential structure of the cotangent bundle of a locally Euclidean differential space.

In the spirit of the theory of differential spaces, the cotangent bundle $T^{*}S$ of a differential space S with differential structure $C^{\infty }(S)$ is defined as the set differentials $\mathrm{d}{f}_{\mid x}$ of functions $f\in C^{\infty }(S)$ at points $x\in S$ . First, we determine the differential structure $C^{\infty }(T^{*}W)$ of the cotangent bundle $T^{*}W$ of a differential subspace W of ${\mathrm{ℝ}}^n$ . We extend this result to define differential structure $C^{\infty }(T^{*}S)$ of the cotangent bundle $T^{*}S$ of a locally Euclidean differential space S and prove that this differential structure is locally Euclidean. Next, we derive the canonical 1-form ${\theta }_S$ of the cotangent bundle $T^{*}S$ of S. If the canonical 2-form ${ɷ}_S=\mathrm{d}{\theta }_S$ is non-degenerate, then it defines a symplectic structure of $T^{*}S$ . We show that the partition $\mathfrak{M}(S)$ of S induces a partition $\{T_M^{*}S\mid M\in$ $\mathfrak{M}(S)\}$ of $T^{*}S$ by submanifolds, where $T_M^{*}S={\pi }_S^{-1}(M)$ and ${\pi }_S:T^{*}S\to S$ are the cotangent bundle projections.

2. Differential spaces

Definition 2.1. A differential structure on a topological space S is a family $C^{\infty }(S)$ of real-valued functions on S that satisfy the following conditions:

1. The family

   $$\{f^{-1}(I)\mid f\in C^{\infty }(S)\mathrm{and}I\mathrm{is}\mathrm{an}\mathrm{open}\mathrm{interval}\mathrm{in}\mathrm{ℝ}\}$$

   is a sub-basis of the topology of S.

2. If $f_1,\dots ,f_n\in C^{\infty }(S)$ and $F\in C^{\infty }({\mathrm{ℝ}}^n)$ ² , then $F(f_1,\dots ,f_n)\in C^{\infty }(S)$ .

3. A function $f:S\to \mathrm{ℝ}$ is in $C^{\infty }(S)$ if, for each $x\in S$ , there is an open neighbourhood V of x in S and a function $f_x\in C^{\infty }(S)$ satisfying

   $$f_{x\mid V}=f_{\mid V}.$$

A topological space S endowed with a differential structure $C^{\infty }(S)$ is called a differential space. Functions $f\in C^{\infty }(S)$ are referred to as smooth functions on S. A map $\phi :S\to R$ between differential spaces with differential structures $C^{\infty }(S)$ and $C^{\infty }(R)$ , respectively, is smooth if, for every $f\in C^{\infty }(R)$ , ${\phi }^{*}f\in C^{\infty }(S)$ , where ${\phi }^{*}f=f\circ \phi$ is the pull-back of f by φ. A smooth map $\phi :S\to R$ is a diffeomorphism if it is invertible and its inverse ${\phi }^{-1}:R\to S$ is smooth. If a map $\phi :S\to R$ is smooth, then it is continuous.

A simple way of defining a differential structure on a set S is as follows.

Definition 2.2. Choose a family of functions $\mathcal{F}$ on S. Endow S with the topology generated by a sub-basis

$$\{f^{-1}(I)\mid f\in \mathcal{F}\mathrm{and}I\mathrm{is}\mathrm{an}\mathrm{open}\mathrm{interval}\mathrm{in}\mathrm{ℝ}\}.$$

(1)

The differential structure $C^{\infty }(S)$ generated by $\mathcal{F}$ consists of functions $h:S\to \mathrm{ℝ}$ , such that, for each $x\in S$ , there exist an open neighbourhood V of x, an integer $n\in \mathrm{\mathbb{N}},$ functions $f_1,\dots ,f_n\in \mathcal{F}$ , and $F\in C^{\infty }({\mathrm{ℝ}}^n)$ such that

$$h_{\mid V}=F{(f_1,\dots ,f_n)}_{\mid V}.$$

(2)

A proof that the differential structure $C^{\infty }(S)$ generated by $\mathcal{F}$ in Definition 2.2 satisfies all conditions of Definition 2.1 can be found in Śniatycki [13].

Definition 2.3. Let S be a differential space with a differential structure $C^{\infty }(S)$ . For every subset $R$ of S, let $C^{\infty }(R)$ be the differential structure generated by the family

$$\mathcal{R}(R)=\{F_{\mid R}\mid F\in C^{\infty }(S)\}$$

of restrictions to R of smooth functions on S. The resulting differential space R is called a differential subspace of S. In this differential structure, the inclusion map ${\iota }_R:R\to S$ is smooth.

We may extend the notion of a differential subspace of a differential space as follows.

Definition 2.4. Let $\kappa :R\to S$ be a one-to-one map of a set R into a differential space S. It defines a bijection $R\to \kappa (R)\subseteq S$ . Let $C^{\infty }(\kappa (R))$ be the differential structure of $\kappa (R)$ generated by restrictions to $\kappa (R)$ of smooth functions on S. It is easy to verify that

$$C^{\infty }(R)=\{{\kappa }^{*}f=f\circ \kappa :R\to \mathrm{ℝ}\mid f\in C^{\infty }(\kappa (R))\}$$

is a differential structure on R such that $\kappa :R\to S$ is smooth. We say $C^{\infty }(R)$ is the subspace differential structure of R induced by $\phi :R\to S$ .

Definition 2.5. Let $(S,C^{\infty }(S))$ be a differential space. An equivalence relation ≈ on S defines a subset R of $S\times S$ such that, if $(x,y)\in S\times S$ , then

$$(x,y)\in R⟺x\simeq y.$$

For each $x\in S$ we denote by $\left[x\right]$ the ≃ equivalence class x. Let $Q=S∕\approx$ be the set of equivalence classes of the relation ≃ in S, and let $\varrho :S\to Q$ be the quotient projection map, given by $\varrho (x)=\left[x\right]$ for every $x\in S$ . The quotient differential structure of Q is

$$C^{\infty }(Q)=\{f:Q\to \mathrm{ℝ}\mid {\varrho }^{*}f=f\circ \varrho \in C^{\infty }(S)\}.$$

(3)

It is easy to verify that $C^{\infty }(Q)$ given by equation (3) satisfies Definition 2.1. In the quotient differential structure, the projection map $\varrho :S\to Q$ is smooth. It should be noted that the topology of Q defined by the differential structure $C^{\infty }(Q)$ need not coincide with the quotient topology of Q.

We may extend the notion of a quotient differential structure as follows.

Definition 2.6. Let $(S,C^{\infty }(S))$ be a differential space. A map $\phi :S\to R$ of S onto a set R defines an equivalence relation ≈ on S as follows:

$$x\approx y⟺\phi (x)=\phi (y).$$

We have a natural identification of the quotient $Q=S∕\approx \to R:\left[x\right]\mapsto \phi (x)$ . With this identification, the quotient differential structure $C^{\infty }(Q)$ defines a quotient differential structure $C^{\infty }(R)$ induced by $\phi :S\to R$ .

Proposition 2.7. A smooth map $\phi :S\to P$ between differential spaces S and P may be written as $\phi =\iota \circ \lambda$ , where $\iota :R=\phi (S)\to P$ is the inclusion of the range $R=\phi (S)\subseteq P$ into P and $\lambda :S\to R$ is the restriction of $\phi :S\to P$ to codomain R which is smooth. Moreover, λ maps $S$ onto R and $\iota :R\to S$ is a one-to-one map.

Proof. The inclusion $\iota :R\to S$ induces differential structure $C^{\infty }(R)$ on R, generated by restrictions to R of smooth functions on P. A set $V\subseteq R$ is open if and only if there exists an open set U in P such that $V=R\cap U$ . ³ Hence, for every open set V in R,

$${\lambda }^{-1}(V)={\lambda }^{-1}(R\cap U)={\lambda }^{-1}({\iota }^{-1}(U))={(\iota \circ \lambda )}^{-1}(U)={\phi }^{-1}(U)$$

is open in S, because φ is smooth. Therefore, λ is continuous.

We want to show that $\lambda :S\to R$ is smooth. Let $f:R\to \mathrm{ℝ}$ be a function on R. If $f\in C^{\infty }(R)$ , then for every point $y\in R\subseteq P$ , there exists an open neighbourhood U of y in $R\subseteq P$ and a function $F\in C^{\infty }(P)$ such that $f_{\mid V}=F_{\mid V}={(F_{\mid R})}_{\mid V}=F\circ \iota$ . Therefore,

$$\begin{array}{cccc}\multicolumn{1}{c}{{({\lambda }^{*}f)}_{\mid {\lambda }^{-1}(V)}}& \multicolumn{1}{c}{=}& \multicolumn{1}{c}{{(f\circ \lambda )}_{\mid {\lambda }^{-1}(V)}=f_{\mid V}\circ {\lambda }_{\mid {\lambda }^{-1}(V)}=F_{\mid V}\circ {\lambda }_{\mid {\lambda }^{-1}(V)}}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{=}& \multicolumn{1}{c}{{(F\circ \iota \circ \lambda )}_{\mid {\lambda }^{-1}(V)}={(F\circ \phi )}_{\mid {\lambda }^{-1}(V)}.={({\phi }^{*}F)}_{\mid {\lambda }^{-1}(V)}.}& \multicolumn{1}{c}{}\end{array}$$

Since $\lambda :S\to R$ is continuous and $V=U\cap R$ is open in R, the inverse image ${\lambda }^{-1}(V)$ is open in S. Therefore, for every $x\in S$ , there exists an open neighbourhood V of $y=\phi (x)\in R\subseteq P$ and a function $F\in C^{\infty }(P)$ such that ${({\lambda }^{*}f)}_{\mid {\lambda }^{-1}(V)}={({\phi }^{*}F)}_{\mid {\lambda }^{-1}(V)}$ . Therefore, ${\lambda }^{*}f\in C^{\infty }(S)$ , which implies that $\lambda :S\to R$ is smooth in the differential structure of R, induced by its inclusion map $\iota :R\to S$ . ■

Definition 2.8. A differential space S is locally Euclidean if, for every $x\in S$ , there exists a diffeomorphism $\alpha :V_{\alpha }\to W_{\alpha }$ , where $V_{\alpha }$ is an open differential subspace of S containing x, and $W_{\alpha }$ is a differential subspace of some ${\mathrm{ℝ}}^{d_{\alpha }}$ . The collection $\mathfrak{A}=\{\alpha :V_{\alpha }\to W_{\alpha }\}$ is called an atlas on S corresponding to $C^{\infty }(S)$ .

Proposition 2.9. Let S be a locally Euclidean differential space with the corresponding atlas $\mathfrak{A}(S)=\{\alpha :V_{\alpha }\to W_{\alpha }\subseteq {\mathrm{ℝ}}^{d_{\alpha }}\}$ . Then, $f:S\to \mathrm{ℝ}$ is in $C^{\infty }(S)$ if and only if, for every $\alpha \in \mathfrak{A}$ , the restriction $f_{\mid V_{\alpha }}\in C^{\infty }(V_{\alpha })$ .

Proof. Let S be a differential space. Hence, it is endowed with a differential structure $C^{\infty }(S)$ , which leads to the atlas $\mathfrak{A}$ . Since each $V_{\alpha }$ is a differential subspace of S, it follows that its differential structure is generated by restrictions to $V_{\alpha }$ of functions in $C^{\infty }(S)$ . Hence, a function $f:S\to \mathrm{ℝ}$ is in $C^{\infty }(S)$ if, for every $\alpha \in \mathfrak{A}$ , the restriction $f_{\mid V_{\alpha }}\in C^{\infty }(V_{\alpha })$ .

Conversely, let

$$\mathcal{F}=\{f:S\to \mathrm{ℝ}\mid f_{\mid V_{\alpha }}\in C^{\infty }(V_{\alpha })\forall \alpha \in \mathfrak{A}\}.$$

We want to show that $\mathcal{F}$ satisfies the conditions of Definition 2.1.

1. For every $\alpha \in \mathfrak{A}(S)$ , the family $\{f_{\mid V_{\alpha }}^{-1}((a,b))\mid f_{\mid V_{\alpha }}\in C^{\infty }(V_{\alpha })$ , $a,b\in \mathrm{ℝ}$ and $a<b\}$ is a sub-basis of the topology of $V_{\alpha }$ .

   Since $S={\mathrm{\cup }}_{\alpha \in \mathfrak{A}}{V}_{\alpha }$ and each $V_{\alpha }$ is open in S, it follows that $\{f^{-1}(a,b)\mid f\in$ $\mathcal{F}$ , $a,b\in \mathrm{ℝ}$ and $a<b\}$ is a sub-basis of the topology of S.

2. If $f_1,\dots ,f_n\in \mathcal{F}$ and $F\in C^{\infty }({\mathrm{ℝ}}^n)$ , then $F(f_1,\dots ,f_n)\in \mathcal{F}$ is self-evident.

3. This condition is incorporated in the definition of $\mathcal{F}$ . It follows that $\mathcal{F}$ satisfies all conditions of Definition 2.1. Therefore, $C^{\infty }(S)=\mathcal{F}$ is the differential structure of a locally Euclidean space S with atlas $\mathfrak{A}$ . ■

2.1. Derivations and vector fields on locally Euclidean spaces

Let S be a locally Euclidean differential space. A derivation of $C^{\infty }(S)$ is a linear map $X:C^{\infty }(S)\to C^{\infty }(S):f\mapsto \mathrm{Xf}$ satisfying Leibniz’ rule

$$X(f_1{f}_2)=(X{f}_1)f_2+f_1(X{f}_2)$$

(4)

for every $f_1$ , $f_2\in C^{\infty }(S)$ . The space $\mathrm{Der}{C}^{\infty }(S)$ of derivations of $C^{\infty }(S)$ is a Lie algebra with Lie bracket

$$[X_1,X_2]f=X_1(X_{2}f)-X_2(X_{1}f)$$

(5)

for every $X_1$ , $X_2\in \mathrm{Der}{C}^{\infty }(S)$ and every $f\in C^{\infty }(S)$ . In addition, $\mathrm{Der}{C}^{\infty }(S)$ is a module over the ring $C^{\infty }(S)$ with $[f{X}_1,X_2]=f[X_1,X_2]$ and

$$[X_1,f{X}_2]=(X_{1}f)X_2+f[X_1,X_2]$$

(6)

for every $X_1$ , $X_2\in \mathrm{Der}{C}^{\infty }(S)$ and every $f\in C^{\infty }(S)$ .

An integral curve of X originating at $x_0\in S$ is a map $c:I\to S$ , where I is a connected subset of $\mathrm{ℝ}$ containing 0, such that $c(0)=x_0$ and

$$\frac{d}{\mathrm{dt}}f(c(t))=(\mathrm{Xf})(c(t))\mathrm{for}\mathrm{every}f\in C^{\infty }(S)\mathrm{and}\mathrm{every}t\in I,$$

(7)

whenever the interior of I is not empty.

Theorem 2.10. Let S be a locally Euclidean differential space and let X be a derivation of $C^{\infty }(S)$ . For every $x\in S$ , there exists a unique maximal integral curve c of X such that $c(0)=x$ .

Proof. See the proof of Theorem 3.2.1 in the work by Śniatycki [13]. ■

Let X be a derivation of $C^{\infty }(S)$ . We denote by ${\mathrm{e}}^{\mathrm{tX}}(x)$ the point on the maximal integral curve of X, originating at x, corresponding to the value t of the parameter. Given $x\in S$ , ${\mathrm{e}}^{\mathrm{tX}}(x)$ is defined for t in an interval $I_x$ containing zero, and ${\mathrm{e}}^{0X}(x)(x)=x$ . If t, s, and $t+s$ are in $I_x$ , $s\in I_{{\mathrm{e}}^{\mathrm{tX}}(x)}$ , and $t\in I_{{\mathrm{e}}^{\mathrm{sX}}(x)}$ , then,

$${\mathrm{e}}^{(s+t)X}(x)={\mathrm{e}}^{\mathrm{sX}}({\mathrm{e}}^{\mathrm{tX}}(x))={\mathrm{e}}^{\mathrm{tX}}(x)({\mathrm{e}}^{\mathrm{sX}}(x)).$$

In the case when S is a manifold, the map ${\mathrm{e}}^{\mathrm{tX}}$ is a local one-parameter group of local diffeomorphisms of S. For a differential space S, ${\mathrm{e}}^{\mathrm{tX}}:x\mapsto {\mathrm{e}}^{\mathrm{tX}}(x)$ might fail to be a local diffeomorphism.

A vector field on a locally Euclidean differential space S is a derivation X of $C^{\infty }(S)$ if ${\mathrm{e}}^{\mathrm{tX}}$ is a local one-parameter group of local diffeomorphisms of S. We denote by $\mathfrak{X}(S)$ the family of all vector fields on S. It is a Lie subalgebra of $\mathrm{Der}{C}^{\infty }(S)$ invariant under multiplication by smooth functions [9,12].

The orbit through $x_0$ of the family $\mathfrak{X}(S)$ of vector fields on S is the set M of points x in S that can be joined to $x_0$ by a piecewise smooth integral curve of vector fields in $\mathfrak{X}(S)$ ;

$$M=\{{\mathrm{e}}^{t_n{X}_n}\circ {\mathrm{e}}^{t_{n-1}{X}_{n-1}}\circ \dots .\circ {\mathrm{e}}^{t_1{X}_1}(x_0)\mid X_1,\dots ,X_n\in \mathfrak{X}(S)t_1,\dots ,t_n\in \mathrm{ℝ},n\in \mathrm{\mathbb{N}}\}.$$

Theorem 2.11. Orbits M of the family $\mathfrak{X}(S)$ of vector fields on a locally Euclidean differential space S are submanifolds of S. In the manifold topology of M, the differential structure on M induced by its inclusion in S coincides with its manifold differential structure.

Proof. See the work by Śniatycki [14], or the proof of Theorem 3.4.5 in the work by Śniatycki [13]. ■

The collection $\mathfrak{M}(S)$ of all orbits of $\mathfrak{X}(S)$ forms a partition of S by smooth manifolds, which satisfies the following condition, ⁴

Frontier Condition 2.12. For $M,M^{\prime }\in \mathfrak{M}(S)$ , if $M^{\prime }\cap \overline{M}\ne \varnothing$ , either $M^{\prime }=M$ or $M^{\prime }=\overline{M}\setminus M$ .

If $\mathfrak{M}(S)$ is locally finite, then it is a stratification of S. The partition $\mathfrak{M}(S)$ of a differential space S by smooth manifolds indicates aspects of the geometry of S which can be successfully studied by the methods of differential geometry.

3. Vectors and covectors

In preparation for the analysis of the differential structure of the cotangent bundle of a locally Euclidean differential space, we discuss vectors and covectors as concrete objects on a differential space and describe their properties, which do not depend on the choice of the differential structures of the tangent or the cotangent bundle.

Definition 3.1. Vectors tangent to a differential space S at a point $x\in S$ are derivations of $C^{\infty }(S)$ at x, that is, linear maps ${\mathrm{v}}_x:C^{\infty }(S)\to \mathrm{ℝ}:f\mapsto {\mathrm{v}}_{x}f$ that satisfy Leibniz’ rule:

$${\mathrm{v}}_x(f_1{f}_2)=({\mathrm{v}}_x{f}_1)f_2(x)+f_1(x)({\mathrm{v}}_x{f}_2).$$

The tangent space to S at a point x is the space $T_{x}S$ of all derivations ${\mathrm{v}}_x:C^{\infty }(S)\to \mathrm{ℝ}$ at x. The tangent bundle of S is as follows:

$$\mathrm{TS}=\coprod _{x\in S}{T}_{x}S.$$

(8)

The tangent bundle projection is the map ${\tau }_S:\mathrm{TS}\to S:{\mathrm{v}}_x\mapsto \tau ({\mathrm{v}}_x)=x$ .

Since the point x of attachment of a vector ${\mathrm{v}}_x\in \mathrm{TS}$ is determined by the tangent bundle projection ${\tau }_S:\mathrm{TS}\to S$ , we may write ${\mathrm{v}}_x=\mathrm{v}$ and $x=\tau (\mathrm{v})$ . For V $\subseteq S$ , let

$$T_{V}S=\coprod _{x\in V}{T}_{x}S={\tau }_S^{-1}(V)\subseteq \mathrm{TS}.$$

(9)

Let S be a locally Euclidean differential space. Then, its tangent bundle TS is also locally Euclidean. A smooth map $\sigma :S\to \mathrm{TS}$ such that ${\tau }_S\circ \sigma ={\mathrm{id}}_S$ is called a section of TS.

The differential of a function $f\in C^{\infty }(S)$ is a linear map:

$$\mathrm{d}f:\mathrm{TS}\to \mathrm{ℝ}:\mathrm{v}\mapsto \mathrm{d}f(\mathrm{v})=\mathrm{v}f.$$

For every $x\in S$ , we denote by $\mathrm{d}{f}_{\mid x}$ the restriction of $\mathrm{d}f$ to $T_{x}S$ and use the notation

$$\mathrm{d}{f}_{\mid x}:T_{x}S\to \mathrm{ℝ}:\mathrm{v}\mapsto \mathrm{d}{f}_{\mid x}(\mathrm{v})=⟨\mathrm{d}{f}_{\mid x}\mid \mathrm{v}⟩=⟨\mathrm{d}f\mid \mathrm{v}⟩=\mathrm{v}f.$$

(10)

Definition 3.2. Covectors at $x\in S$ are differentials $\mathrm{d}{f}_{\mid x}$ at x of functions $f\in C^{\infty }(S)$ . The space of covectors at $x\in S$ is called the cotangent space of S at x, and it is denoted $T_x^{*}S$ . Thus,

$$T_x^{*}S=\{\mathrm{d}{f}_{\mid x}\mid f\in C^{\infty }(S)\}.$$

(11)

The cotangent bundle space of S is the space

$$T^{*}S=\coprod _{x\in S}{T}_x^{*}S=\{\mathrm{d}{f}_{\mid x}\mid x\in S,f\in C^{\infty }(S)\}$$

(12)

of all covectors at all points of S. The cotangent bundle projection is the map ${\pi }_S:T^{*}S\to S$ assigning to each $\mathrm{d}{f}_{\mid x}\in T^{*}S$ the point $x\in S$ at which the differential $\mathrm{d}f$ is evaluated,

$${\pi }_S(\mathrm{d}{f}_{\mid x})=x.$$

(13)

For V $\subseteq S$ ,

$$T_V^{*}S=\coprod _{x\in V}{T}_x^{*}S={\pi }_S^{-1}(V)\subseteq T^{*}S.$$

(14)

Equation (10) can be interpreted as the evaluation function on the fibre product $T^{*}S{\times }_S\mathrm{TS}$ ,

$$⟨\cdot \mid \cdot ⟩:T^{*}S{\times }_S\mathrm{TS}\to \mathrm{ℝ}:(\mathrm{d}{f}_{\mid x},{\mathrm{v}}_x)\mapsto ⟨\mathrm{d}{f}_{\mid x}\mid {\mathrm{v}}_x⟩={\mathrm{v}}_{x}f.$$

(15)

Since $C^{\infty }(S)$ is closed under the operations of addition and multiplication by constants, and the derivation in $T_{x}S$ is linear, it follows that $T_x^{*}S$ is closed under addition and multiplication by constants. For every $\mathrm{d}{f}_{1\mid x}$ and $\mathrm{d}{f}_{2\mid x}$ in $T_x^{*}S$ and every $a_1,a_2\in \mathrm{ℝ}$ ,

$$a_1\mathrm{d}{f}_{1\mid x}+a_2\mathrm{d}{f}_{2\mid x}=\mathrm{d}{(a_1{f}_1+a_2{f}_2)}_{\mid x}\in T_x^{*}S.$$

(16)

Hence, $T_x^{*}S$ is a vector space for every $x\in S$ .

Proposition 3.3. Let S be a locally Euclidean differential space. Then, $\dim T_x^{*}S=\dim T_{x}S$ for every $x\in S$ . Moreover,

$$\mathrm{d}{f}_{\mid x}=0⟺⟨\mathrm{d}{f}_{\mid x}\mid \mathrm{v}⟩=0\forall \mathrm{v}\in T_{x}S.$$

Proof. Since S is locally Euclidean, it follows that $\dim \mathrm{TS}$ is locally finite. Suppose that $\dim T_{x}S=n$ . That is, there exist n linearly independent derivations ${\mathrm{v}}_1,\dots ,{\mathrm{v}}_n$ of $C^{\infty }(S)$ at $x\in S$ . Suppose that $\dim T_x^{*}S=m$ . Then there exist m linearly independent differentials $\mathrm{d}{f}_{1\mid x},\dots ,\mathrm{d}{f}_{m\mid x}$ . Suppose that $m<n$ . It implies that, for every function $f\in C^{\infty }(S)$ , there exist constants $a_1,\dots ,a_m$ such that

$$\mathrm{d}{f}_{\mid x}=a_1\mathrm{d}{f}_{1\mid x}+\cdots +a_m\mathrm{d}{f}_{m\mid x}=\mathrm{d}{(a_1{f}_1+\cdots +a_m{f}_m)}_{\mid x}.$$

The second equality above is valid because $a_1,\dots ,a_m$ is constant. Therefore, for every $\mathrm{v}\in T_{x}S$ ,

$$⟨\mathrm{d}{f}_{\mid x}\mid \mathrm{v}⟩=\sum _{i=1}^m{a}_i⟨\mathrm{d}{f}_{i\mid x}\mid \mathrm{v}⟩.$$

and $⟨\mathrm{d}{f}_{\mid x}\mid \mathrm{v}⟩=0$ if and only if $⟨\mathrm{d}{f}_{i\mid x}\mid \mathrm{v}⟩=0$ for all $i=1,\dots ,m$ . This implies that $T_{x}S$ is spanned by m independent derivations of $C^{\infty }(S)$ at x and contradicts the supposition that there exists $n>m$ linearly independent derivations ${\mathrm{v}}_1,\dots ,{\mathrm{v}}_n$ of $C^{\infty }(S)$ at $x\in S$ . In a similar way, we may obtain a contradiction with a supposition that $n<m$ . Therefore, $m=n$ and $\dim T_{x}S=\dim T_x^{*}S$ . In particular, the dimension of $T^{*}S$ is locally finite. ■

A smooth map $\phi :S\to R$ gives rise to the tangent map

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{T\phi }:\mathrm{TS}\to \mathrm{TR}:\mathrm{v}\mapsto \mathrm{T\phi }(\mathrm{v})\mathrm{such}\mathrm{that}}\\ \multicolumn{1}{c}{(\mathrm{T\phi }(\mathrm{v}))f=\mathrm{v}(f\circ \phi )=\mathrm{v}({\phi }^{*}f)\mathrm{for}\mathrm{every}f\in C^{\infty }(R),}& \multicolumn{1}{c}{}\end{array}$$

(17)

which is smooth and the following diagram commutes

$$\begin{array}{ccccc}\multicolumn{1}{c}{}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{\mathrm{T\phi }}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{\mathrm{TS}}& \multicolumn{1}{c}{\to }& \multicolumn{1}{c}{\mathrm{TR}}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{{\tau }_S}& \multicolumn{1}{c}{\downarrow }& \multicolumn{1}{c}{}& \multicolumn{1}{c}{\downarrow }& \multicolumn{1}{c}{{\tau }_R}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{S}& \multicolumn{1}{c}{\to }& \multicolumn{1}{c}{R}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{\phi }& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}\end{array},$$

where ${\tau }_S:\mathrm{TS}\to S$ and ${\tau }_R:\mathrm{TR}\to R$ are the tangent bundle projections, see Śniatycki [13].

For every $x\in S$ , the restriction of $\mathrm{T\phi }$ to $T_{x}S$ is a linear map $T{\phi }_x:T_{x}S\to T_{y}R$ , where $y=\phi (x)$ . Moreover, for every ${\mathrm{v}}_x\in T_{x}S$ and $f\in C^{\infty }(R),T{\phi }_x({\mathrm{v}}_x)\in T_{y}R$ and ${\phi }^{*}f=f\circ \phi \in C^{\infty }(S)$ because $\phi :S\to R$ is smooth. Hence,

$$⟨\mathrm{d}{({\phi }^{*}f)}_{\mid x}\mid \mathrm{v}⟩=\mathrm{v}({\phi }^{*}f)=T{\phi }_x(\mathrm{v})f=⟨\mathrm{d}{f}_{\mid y}\mid T{\phi }_x(\mathrm{v})⟩$$

(18)

This equation describes a covector $\mathrm{d}{({\phi }^{*}f)}_{\mid x}\in T_x^{*}S$ acting on an arbitrary vector $\mathrm{v}\in T_{x}S$ in terms of a covector $\mathrm{d}{f}_{\mid y}\in T_y^{*}R$ acting on $\mathrm{T\phi }(\mathrm{v})$ , where $y=\phi (x)$ . We can rewrite equation (18) in the following form:

$$\mathrm{d}{({\phi }^{*}f)}_{\mid x}=\mathrm{d}{f}_{\mid \phi (x)}\circ \mathrm{T\phi }.$$

(19)

Definition 3.4. A one-to-one smooth map $\phi :S\to R$ gives rise to its cotangent map ⁵

$$\begin{array}{ccc}\multicolumn{1}{c}{T^{*}\phi :T_{\phi (S)}^{*}R\to T^{*}S:\mathrm{d}{f}_{\mid \phi (x)}}& \multicolumn{1}{c}{\mapsto }& \multicolumn{1}{c}{T^{*}\phi (\mathrm{d}{f}_{\mid \phi (x)})=\mathrm{d}{f}_{\mid \phi (x)}\circ \mathrm{T\phi }}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{=}& \multicolumn{1}{c}{\mathrm{d}{(f\circ \phi )}_{\mid x}=\mathrm{d}{({\phi }^{*}f)}_{\mid x}.}& \multicolumn{1}{c}{}\end{array}$$

(20)

Lemma 3.5. Let V be a differential subspace of S with the inclusion map ${\iota }_V:V\to S$ . If V is open in S, then $T^{*}V=T_V^{*}S={\pi }_S^{-1}(V)\subseteq T^{*}S$ and $T^{*}{\iota }_V:T_V^{*}S\to T^{*}V:\mathrm{d}{f}_{\mid x}\mapsto \mathrm{d}{f}_{\mid x}$ is the identity map.

Proof. By definition, $T_V^{*}S={\pi }_S^{-1}(V)=\{\mathrm{d}{f}_{\mid x}\mid f\in C^{\infty }(S)$ , $x\in V\}$ and $T^{*}V=\{\mathrm{d}{h}_{\mid x}\mid h\in C^{\infty }(V)$ , $x\in V\}$ , where $C^{\infty }(V)$ is generated by the restrictions to V of functions $f\in C^{\infty }(S)$ . Definition 2.2 ensures that, for every $x\in V$ and every $h\in C^{\infty }(V)$ , there exists an open neighbourhood U of x in V, an integer $n\in \mathrm{\mathbb{N}}$ , functions $f_1,\dots ,f_n\in C^{\infty }(S)$ and $F\in C^{\infty }({\mathrm{ℝ}}^n)$ such that $h_{\mid U}=F{(f_1,\dots ,f_n)}_{\mid U}$ . Then,

$$\mathrm{d}{h}_{\mid x}=\mathrm{d}{(h_{\mid U})}_{\mid x}=\mathrm{d}{(F{(f_1,\dots ,f_n)}_{\mid U})}_{\mid x}=\mathrm{d}F{(f_1,\dots ,f)}_{\mid x}.$$

But $F(f_1,\dots ,f)\in C^{\infty }(S)$ , which implies that $\mathrm{d}{h}_{\mid x}\in T_V^{*}S$ . Hence, $T_V^{*}S=T^{*}V$ .

Since ${\iota }_V:V\to S:x\mapsto {\iota }_V(x)=x$ is the restriction to $V\subseteq S$ of the identity in S, equation (20) yields

$$T^{*}{\iota }_V:T_V^{*}S\to T^{*}V:\mathrm{d}{f}_x=\mathrm{d}{f}_{\mid {\iota }_V(x)}\mapsto \mathrm{d}{f}_{\mid x}.$$

This completes the proof. ■

For $X\in \mathrm{Der}{C}^{\infty }(S)$ , and $x\in S$ , the evaluation of X at x is a vector $X(x)\in T_{x}S$ such that, for every $f\in C^{\infty }(S)$ ,

$$X(x)f=(\mathrm{Xf})(x).$$

The map $S\to \mathrm{TS}:x\mapsto X(x)$ is a smooth section of ${\tau }_S:\mathrm{TS}\to S$ .

Proposition 3.6. A section $\sigma :S\to \mathrm{TS}$ of a locally Euclidean space S determines a derivation, also denoted σ, such that $(\mathrm{\sigma f})(x)=\sigma (x)f$ for every $f\in C^{\infty }(S)$ and every $x\in S$ .

Proof. In order to prove the statement of the proposition, we assume first that S is a differential subspace ${\mathrm{ℝ}}^n$ with the inclusion map ${\iota }_S:S\to {\mathrm{ℝ}}^n$ . Then, $C^{\infty }(S)$ is generated by restrictions to S of functions in $C^{\infty }({\mathrm{ℝ}}^n)$ . But $C^{\infty }({\mathrm{ℝ}}^n)$ is generated by the Cartesian coordinate functions $x^1,\dots ,x^n:{\mathrm{ℝ}}^n\to \mathrm{ℝ}$ . For every $i=1,\dots ,n$ , and every $\mathit{x}=(x^1,\dots ,x^i,\dots ,x^n)\in {\mathrm{ℝ}}^n$ ,

$$x^i(\mathit{x})=x^i(x^1,\dots ,x^i,\dots ,x^n)=x^i,$$

where on the right-hand side is the value of the coordinate $x^i$ at the point $\mathit{x}$ . The tangent bundle TS is a differential subspace of ${\mathrm{ℝ}}^{2n}$ with inclusion map $T_{\mathrm{TS}}=T{\iota }_S$ and $C^{\infty }(\mathrm{TS})$ is generated by restrictions to TS of functions in $C^{\infty }(T{\mathrm{ℝ}}^n)$ . But $C^{\infty }(T{\mathrm{ℝ}}^n)$ is generated by the coordinates

$$\{q^1,\dots ,q^n,v^1,\dots ,v^n\mid C^{\infty }({\mathrm{ℝ}}^n)\}$$

in ${\mathrm{ℝ}}^{2n}$ , such that, for every ${\mathrm{v}}_x\in T{\mathrm{ℝ}}^n$ , and every $i=1,\dots ,n$ ,

$$q^i({\mathrm{v}}_{\mathit{x}})={\tau }_{{\mathrm{ℝ}}^n}^{*}{x}^i({\mathrm{v}}_{\mathit{x}})=x^i({\tau }_{{\mathrm{ℝ}}^n}({\mathrm{v}}_{\mathit{x}}))=x^i(\mathit{x})=x^i,\mathrm{and}{v}^i({\mathrm{v}}_{\mathit{x}})={\mathrm{v}}_{\mathit{x}}{x}^i,$$

where ${\tau }_{{\mathrm{ℝ}}^n}:T{\mathrm{ℝ}}^n\to {\mathrm{ℝ}}^n$ is the tangent bundle projection and ${\mathrm{v}}_{\mathit{x}}{x}^i$ is the action of the derivation ${\mathrm{v}}_{\mathit{x}}$ at $\mathit{x}$ on the coordinate function $x^i$ . The coordinates $x^i$ restrict to $x_{\mid S}^i\in C^{\infty }(S)$ , while the coordinates $q^i$ and $v^i$ restrict to $q_{\mid \mathrm{TS}}^i,v_{\mid \mathrm{TS}}^i\in C^{\infty }(\mathrm{TS})$ for $i=1,\dots ,n$ .

If S is closed in ${\mathrm{ℝ}}^n$ , then TS is closed in $T{\mathrm{ℝ}}^n$ and every function $f\in C^{\infty }(S)$ is the restriction to S of a function F in ${\mathrm{ℝ}}^n$ expressed in terms of the coordinates $x^1,\dots ,x^n$ , i.e.,

$$f=F{(x^1,\dots ,x^n)}_{\mid S}=F(x_{\mid S}^1,\dots ,x_{\mid S}^n).$$

Similarly, every function $e\in C^{\infty }(\mathrm{TS})$ is the restriction to S of a function E in $C^{\infty }({\mathrm{ℝ}}^{2n})$ expressed in terms of the coordinates $q^1,\dots ,q^n$ , $v^1,\dots ,v^n$ , i.e.,

$$e=E{(q^1,\dots ,q^n,v^1,\dots ,v^n)}_{\mid S}=E(q_{\mid \mathrm{TS}}^1,\dots ,q_{\mid \mathrm{TS}}^n,v_{\mid \mathrm{TS}}^1,\dots ,v_{\mid \mathrm{TS}}^n).$$

(21)

To verify the statement of the proposition, it suffices to observe that, for every $f\in C^{\infty }(S)$ , the function $\mathrm{\sigma f}:\mathit{x}\mapsto$ $\sigma (\mathit{x})f$ is in $C^{\infty }(S)$ . Since $\sigma :S\to \mathrm{TS}$ is smooth, for every $E\in C^{\infty }({\mathrm{ℝ}}^{2n})$ , ${\sigma }^{*}{E}_{\mid \mathrm{TS}}\in C^{\infty }(S)$ . For $F\in C^{\infty }({\mathrm{ℝ}}^n)$ such that $f=F_{\mid S}=F(x_{\mid S}^1,\dots ,x_{\mid S}^n)$ , and every $\mathit{x}\in S$ ,

$$(\mathrm{\sigma f})(\mathit{x})=\sigma (\mathit{x})f=\sigma (\mathit{x})F(x_{\mid S}^1,\dots ,x_{\mid S}^n)=\sum _{i=1}^n(\frac{\mathrm{\partial F}}{\partial x^i}\sigma (\mathit{x})x^i)=H\circ \sigma (\mathit{x}),$$

where $H={\sum }_{i=1}^n(\frac{\mathrm{\partial F}}{\partial x^i}{v}^i)\in C^{\infty }({\mathrm{ℝ}}^{2n})$ . Hence, $\mathrm{\sigma f}=H\circ \sigma$ $\in C^{\infty }(S)$ because $\sigma :S\to \mathrm{TS}$ and $H:\mathrm{TS}\to R$ are smooth. Thus, a section $\sigma :S\to \mathrm{TS}$ is equivalent to a derivation $\sigma :C^{\infty }(S)\to C^{\infty }(S)$ . ■

For every differential space S, the evaluation map, defined in equation (15), allows derivations $X\in \mathrm{Der}{\mathcal{C}}^{\infty }(S)$ to be interpreted as functions, also denoted X,

$$X:T^{*}S\to \mathrm{ℝ}:\mathrm{d}{f}_{\mid x}\mapsto X(\mathrm{d}{f}_{\mid x})=⟨\mathrm{d}{f}_{\mid x}\mid X(x)⟩=(\mathrm{Xf})(x).$$

(22)

Proposition 3.7. If the tangent bundle TS of a differential space S is spanned by derivations in $\mathrm{Der}{C}^{\infty }(S)$ , then the family

$$\mathcal{F}(T^{*}S)=\{f\circ {\pi }_S,X\mid f\in C^{\infty }(S),\mathrm{and}X\in \mathrm{Der}{C}^{\infty }(S)\},$$

(23)

of functions on $T^{*}S$ separates points in TS and generates a locally Hausdorff, locally Euclidean differential structure $C^{\infty }(T^{*}S)$ of $T^{*}S$ .

Proof. The differential structure $C^{\infty }(S)$ separates points of S. Hence, the family $\{{\pi }_S^{*}f\mid f\in C^{\infty }(S)\}$ separates fibres of the tangent bundle projection. The assumption that TS is spanned by derivations in $\mathrm{Der}{C}^{\infty }(S)$ implies that, for every $x\in S$ , covectors in $T_x^{*}S$ are separated by functions corresponding to derivations $X\in \mathrm{Der}{C}^{\infty }(S)$ . Hence, the differential structure $C^{\infty }(T^{*}S)$ generated by $\mathcal{F}(T^{*}S)$ is Hausdorff. ■

Among subcartesian spaces only regular spaces satisfy the condition that TS is locally spanned by derivations in $\mathrm{Der}{C}^{\infty }(S)$ [15,16]. In the next section, we derive a differential structure of the cotangent bundle of a locally Euclidean differential space S from its atlas $\mathfrak{A}(S)$ .

4. Differential structure of $T^{*}S$

We begin with a review of the standard differential structure of $T^{*}{\mathrm{ℝ}}^d$ from the point of view of the future generalization to differential spaces.

4.1. Differential space perspective at $T^{*}{\mathrm{ℝ}}^d$

The differential structure of ${\mathrm{ℝ}}^d$ is generated by the Cartesian coordinate functions $x^1:{\mathrm{ℝ}}^d\to \mathrm{ℝ},\dots ,$ , $x^d:{\mathrm{ℝ}}^d\to \mathrm{ℝ}$ . In differential geometry one commonly uses the same symbol for the coordinate functions and their values. We follow this notational ambiguity. For each $\mathit{x}\in {\mathrm{ℝ}}^n$ , we write

$$\mathit{x}=(x^1(\mathit{x}),\dots ,x^d(\mathit{x}))=(x^1,\dots ,x^d).$$

(24)

The cotangent bundle of ${\mathrm{ℝ}}^n$ consists of differentials of functions in $C^{\infty }({\mathrm{ℝ}}^n)$ at points in ${\mathrm{ℝ}}^n$ ,

$$T^{*}{\mathrm{ℝ}}^n=\{\mathrm{d}{F}_{\mid \mathit{x}}\mid F\in C^{\infty }({\mathrm{ℝ}}^n)\mathrm{and}\mathit{x}\in {\mathrm{ℝ}}^n\},$$

and the cotangent bundle projection is

$${\pi }_{{\mathrm{ℝ}}^n}:T^{*}{\mathrm{ℝ}}^n\to {\mathrm{ℝ}}^n:\mathrm{d}{F}_{\mid \mathit{x}}\mapsto \mathit{x}.$$

The differential structure of $C^{\infty }(T^{*}{\mathrm{ℝ}}^n)$ is generated by

$$\{{\tau }_{{\mathrm{ℝ}}^n}^{*}F,X\mid F\in C^{\infty }({\mathrm{ℝ}}^n),X\in \mathrm{Der}{C}^{\infty }({\mathrm{ℝ}}^n)\},$$

where ${\tau }_{{\mathrm{ℝ}}^n}^{*}:T{\mathrm{ℝ}}^n\to {\mathrm{ℝ}}^n$ is the tangent bundle projection, and

$$X(\mathrm{d}{F}_{\mid \mathit{x}})=⟨\mathrm{d}{F}_{\mid \mathit{x}}\mid X⟩=(\mathrm{XF})(\mathit{x})$$

for every $F\in C^{\infty }({\mathrm{ℝ}}^n)$ . Therefore, $T^{*}{\mathrm{ℝ}}^n$ is diffeomorphic to ${\mathrm{ℝ}}^{2n}$ . A function $h:T^{*}{\mathrm{ℝ}}^n\to \mathrm{ℝ}$ is in $C^{\infty }(T^{*}{\mathrm{ℝ}}^n)$ if there exists a smooth function $H\in C^{\infty }({\mathrm{ℝ}}^{2n})$ such that

$$h=H({\tau }_{{\mathrm{ℝ}}^n}^{*}{x}^1,\dots ,{\tau }_{{\mathrm{ℝ}}^n}^{*}{x}^n,{\partial }_1,\dots ,{\partial }_n),$$

(25)

where, for $i=1,\dots ,n$ , the derivation ${\partial }_i=\frac{\partial }{\partial x^i}$ is considered to be a function on $T^{*}{\mathrm{ℝ}}^n$ as in equation (22). We can rewrite equation (25) in more familiar coordinate notation as follows. For $i=1,\dots ,n$ , write $q^i=$ ${\tau }_{{\mathrm{ℝ}}^n}^{*}{x}^i$ , and $p_i={\partial }_i\in C^{\infty }({\mathrm{ℝ}}^n)$ . Then,

$$\mathrm{d}{F}_{\mid \mathit{x}}=\sum _{i=1}^n{p}_i(\mathit{x})\mathrm{d}{q}_{\mid \mathit{x}}^i,$$

(26)

for every $F\in C^{\infty }({\mathrm{ℝ}}^n)$ , where $p_i=\frac{\partial }{\partial q^i}F(q^1,\dots ,q^n)$ for $i=1,\dots ,n$ . Also, a function $h:T^{*}{\mathrm{ℝ}}^n\to {\mathrm{ℝ}}^n$ is in $C^{\infty }(T^{*}{\mathrm{ℝ}}^n)$ if there exists a smooth function $H\in C^{\infty }({\mathrm{ℝ}}^{2n})$ such that

$$h=H(q^1,\dots ,q^n,p_1,\dots ,p_n).$$

(27)

A vector ${\mathit{v}}_{\mathit{x}}\in T_{\mathit{x}}{\mathrm{ℝ}}^d$ is a derivation at $\mathit{x}\in {\mathrm{ℝ}}^d$ of $C^{\infty }({\mathrm{ℝ}}^d)$ . That is,

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{v}}_{\mathit{x}}:C^{\infty }({\mathrm{ℝ}}^d)\to {\mathrm{ℝ}}^d:F\mapsto \mathit{v}F\mathrm{such}\mathrm{that}}\\ \multicolumn{1}{c}{{\mathit{v}}_{\mathit{x}}(F_1{F}_2)=({\mathit{v}}_{\mathit{x}}{F}_1)F_2(\mathit{x})+F_1(\mathit{x}){\mathit{v}}_{\mathit{x}}{F}_2.}& \multicolumn{1}{c}{}\end{array}$$

(28)

The map ${\tau }_{{\mathrm{ℝ}}^d}:T{\mathrm{ℝ}}^d\to {\mathrm{ℝ}}^d:{\mathit{v}}_{\mathit{x}}\mapsto {\tau }_{{\mathrm{ℝ}}^d}({\mathit{v}}_{\mathit{x}})=\mathit{x}$ is the tangent bundle projection. In the following, we may omit the subscript $\mathit{x}$ in ${\mathit{v}}_{\mathit{x}}$ and write $\mathit{v}=(x^1,\dots ,x^d;v^1,\dots ,v^d)$ , where $(x^1,\dots ,x^d)=\mathit{x}$ and $v^i=\mathit{v}{x}^i=⟨\mathrm{d}{x}^i\mid \mathit{v}⟩$ for $i=1,\dots ,d$ . The partial derivations ${\partial }_i=\frac{\partial }{\partial x^i}$ for $i=1,\dots ,n$ , form a frame in $T_{\mathit{x}}{\mathrm{ℝ}}^d$ , and we may write $\mathrm{v}={\sum }_{i=1}^d{v}^i{\partial }_{i\mid \mathit{x}}$ .

The Euclidean metric on ${\mathrm{ℝ}}^d$ is the canonical symmetric bilinear function

$$\begin{array}{ccc}\multicolumn{1}{c}{\mathit{g}:T{\mathrm{ℝ}}^d{\times }_{{\mathrm{ℝ}}^d}T{\mathrm{ℝ}}^d\to \mathrm{ℝ}:({\mathit{u}}_{\mathit{x}},{\mathit{v}}_x)}& \multicolumn{1}{c}{\equiv }& \multicolumn{1}{c}{({(u^1,\dots ,u^d)}_{\mathit{x}},{(v^1,\dots ,v^d)}_{\mathit{x}})}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{\mapsto }& \multicolumn{1}{c}{\mathit{g}({\mathit{u}}_{\mathit{x}},{\mathit{v}}_{\mathit{x}})=\sum _{i=1}^d{u}^i{v}^i.}& \multicolumn{1}{c}{}\end{array}$$

(29)

Since the fibre product $T{\mathrm{ℝ}}^d{\times }_{{\mathrm{ℝ}}^d}T{\mathrm{ℝ}}^d$ is diffeomorphic to ${\mathrm{ℝ}}^{3d}$ , it follows that the Euclidean metric $\mathit{g}:T{\mathrm{ℝ}}^d{\times }_{{\mathrm{ℝ}}^d}T{\mathrm{ℝ}}^d\to \mathrm{ℝ}$ is smooth because $\mathit{g}({\mathit{u}}_{\mathit{x}},{\mathit{v}}_x)$ depends smoothly on $\mathit{x}=(x^1,\dots ,x^n)$ , ${\mathit{u}}_{\mathit{x}}={(u^1,\dots ,u^n)}_{\mathit{x}}$ , and ${\mathit{v}}_{\mathit{x}}={(v^1,\dots ,v^n)}_{\mathit{x}}$ .

The Euclidean metric on ${\mathrm{ℝ}}^d$ gives rise to the musical map

$${\mathit{g}}^{♭}:T{\mathrm{ℝ}}^d\to T^{*}{\mathrm{ℝ}}^d:{\mathit{u}}_{\mathit{x}}\equiv {(u^1,\dots ,u^d)}_{\mathit{x}}\mapsto {\mathit{g}}^{♭}({\mathit{u}}_{\mathit{x}})=p_1\mathrm{d}{x}^1+\cdots +p_d\mathrm{d}{x}^d$$

(30)

such that

$$⟨{\mathit{g}}^{♭}({\mathit{u}}_{\mathit{x}})\mid {\mathit{v}}_x⟩=\mathit{g}({\mathit{u}}_{\mathit{x}},{\mathit{v}}_x)\forall {\mathit{v}}_{\mathit{x}}\in T_{\mathit{x}}{\mathrm{ℝ}}^d.$$

(31)

Comparing equation (29) with equations (30) and (31) yields $p_i=u^i$ for every $i=1,\dots ,d$ . Similarly,

$${\mathit{g}}^{♯}:T^{*}{\mathrm{ℝ}}^d\to T{\mathrm{ℝ}}^d:\mathrm{d}{f}_{\mid \mathit{x}}\mapsto {\mathit{g}}^{♯}(\mathrm{d}{f}_{\mid \mathit{x}})\in T_{\mathit{x}}{\mathrm{ℝ}}^d$$

(32)

such that

$$\mathit{g}({\mathit{g}}^{♯}(\mathrm{d}{f}_{\mid \mathit{x}}),{\mathit{v}}_x)=⟨\mathrm{d}{f}_{\mid \mathit{x}}\mid {\mathit{v}}_x⟩\forall {\mathit{v}}_{\mathit{x}}\in T_{\mathit{x}}{\mathrm{ℝ}}^d.$$

(33)

If $\mathrm{d}{f}_{\mid \mathit{x}}={(p_1\mathrm{d}{x}^1+\cdots +p_d\mathrm{d}{x}^d)}_{\mid \mathit{x}}$ , i.e., $p_i=\frac{\mathrm{\partial f}}{\partial x^i}(x)$ for $i=1,\dots ,d$ , then ${\mathit{g}}^{♯}(\mathrm{d}{f}_{\mid \mathit{x}})={\sum }_{i=1}^d{p}_i{\partial }_{i\mid \mathit{x}}$ .

4.2. Vector subspace $\mathit{S}\subseteq {\mathrm{ℝ}}^d$

Let $\mathit{S}\subseteq {\mathrm{ℝ}}^d$ be an m-dimensional vector subspace of ${\mathrm{ℝ}}^d$ with the inclusion map ${\iota }_{\mathit{S}}:\mathit{S}\to {\mathrm{ℝ}}^d$ . The derived map $T{\iota }_{\mathit{S}}:T\mathit{S}\to T{\mathrm{ℝ}}^d$ is the inclusion map of the tangent bundle $T\mathit{S}$ of $\mathit{S}$ into $T{\mathrm{ℝ}}^d$ . We denote by $T_{\mathit{S}}^{*}{\mathrm{ℝ}}^d$ the restriction of $T{\mathrm{ℝ}}^d$ to the points in $\mathit{S}$ , i.e., $T_{\mathit{S}}^{*}{\mathrm{ℝ}}^d={\pi }_{{\mathrm{ℝ}}^d}^{-1}(\mathit{S})$ . The cotangent map of the inclusion map ${\iota }_{\mathit{S}}:\mathit{S}\to {\mathrm{ℝ}}^d$

$$T^{*}{\iota }_{\mathit{S}}:T_{\mathit{S}}^{*}{\mathrm{ℝ}}^d\to T^{*}\mathit{S}:\mathrm{d}{F}_{\mid \mathit{x}}\mapsto \mathrm{d}{({\iota }_{\mathit{S}}^{*}F)}_{\mid \mathit{x}},$$

(34)

maps $T_{\mathit{S}}^{*}{\mathrm{ℝ}}^d$ onto $T^{*}\mathit{S}$ . Hence, it gives rise to an equivalence relation in $T_{\mathit{S}}^{*}{\mathrm{ℝ}}^d$

$$\begin{array}{ccc}\multicolumn{1}{c}{\mathrm{d}{F}_{\mid \mathit{x}}\approx \mathrm{d}{F}_{\mid {\mathit{x}}^{\prime }}^{\prime }}& \multicolumn{1}{c}{⟺}& \multicolumn{1}{c}{T^{*}{\iota }_{\mathit{S}}(\mathrm{d}{F}_{\mid \mathit{x}})=T^{*}{\iota }_{\mathit{S}}(\mathrm{d}{F}_{\mid {\mathit{x}}^{\prime }}^{\prime })}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{⟺}& \multicolumn{1}{c}{\mathit{x}={\mathit{x}}^{\prime }\mathrm{and}⟨\mathrm{d}{F}_{\mid \mathit{x}}-\mathrm{d}{F}_{\mid \mathit{x}}^{\prime }\mid \mathrm{v}⟩=0\mathrm{for}\mathrm{all}\mathrm{v}\in T_{\mathit{x}}\mathit{S}.}& \multicolumn{1}{c}{}\end{array}$$

(35)

The kernel is this equivalence relation is the set of covectors in $T_{\mathit{S}}^{*}{\mathrm{ℝ}}^d$ , which are equivalent to zero. They form the annihilator $\mathrm{AT}\mathit{S}$ of $T\mathit{S}$ in $T_{\mathit{S}}^{*}{\mathrm{ℝ}}^d$ ,

$$\mathrm{AT}\mathit{S}=\{\mathrm{d}{F}_{\mid \mathit{x}}\in T_{\mathit{S}}^{*}{\mathrm{ℝ}}^d\mid ⟨\mathrm{d}{F}_{\mid \mathit{x}}\mid \mathrm{v}⟩=0\forall \mathrm{v}\in T\mathit{S}\},$$

(36)

and the quotient of $T_{\mathit{S}}^{*}{\mathrm{ℝ}}^d$ by $\mathrm{AT}\mathit{S}$ is isomorphic to the cotangent bundle $T^{*}\mathit{S}$ of $\mathit{S}$ ,

$$T^{*}\mathit{S}\equiv T_{\mathit{S}}^{*}{\mathrm{ℝ}}^d∕\mathrm{AT}\mathit{S}.$$

(37)

We denote by ${\mathit{g}}_{\mid \mathit{S}}$ the restriction of the Euclidean metric $\mathit{g}$ on ${\mathrm{ℝ}}^n$ to points in $\mathit{S}\subseteq {\mathrm{ℝ}}^d$ ,

$${\mathit{g}}_{\mid \mathit{S}}:T_{\mathit{S}}{\mathrm{ℝ}}^d{\times }_{{\mathrm{ℝ}}^n}{T}_{\mathit{S}}{\mathrm{ℝ}}^d\to \mathrm{ℝ}:({\mathit{u}}_{\mathit{x}},{\mathit{v}}_x)\mapsto {\mathit{g}}_{\mid \mathit{S}}({\mathit{u}}_{\mathit{x}},{\mathit{v}}_x)=\mathit{g}({\mathit{u}}_{\mathit{x}},{\mathit{v}}_x).$$

(38)

The restriction ${\mathit{g}}_{\mid T\mathit{S}}^{♭}$ of

$${\mathit{g}}_{\mid \mathit{S}}^{♭}:T_{\mathit{S}}{\mathrm{ℝ}}^d\to T_{\mathit{S}}^{*}{\mathrm{ℝ}}^d:{\mathit{u}}_{\mathit{x}}\equiv {(u^1,\dots ,u^d)}_{\mathit{x}}\mapsto {\mathit{g}}_{\mid S}^{♭}({\mathit{u}}_{\mathit{x}})={\mathit{g}}^{♭}({\mathit{u}}_{\mathit{x}})$$

to $T\mathit{S}\subseteq$ $T_{\mathit{S}}{\mathrm{ℝ}}^d$ gives a linear isomorphism

$${\mathit{g}}_{T\mathit{S}}^{♭}={\mathit{g}}_{\mid T\mathit{S}}^{♭}:T\mathit{S}\to T^{*}\mathit{S}:{\mathrm{u}}_{\mathit{x}}\mapsto {\mathit{g}}_{\mid T\mathit{S}}^{♭}({\mathrm{u}}_{\mathit{x}})$$

(39)

such that

$$⟨{\mathit{g}}_{T\mathit{S}}^{♭}({\mathrm{u}}_{\mathit{x}})\mid {\mathrm{v}}_{\mathit{x}}⟩=\mathit{g}({\mathrm{u}}_{\mathit{x}},{\mathrm{v}}_{\mathit{x}})\forall {\mathrm{v}}_{\mathit{x}}\in T_{\mathit{x}}\mathit{S},$$

(40)

with inverse

$${\mathit{g}}_{T^{*}\mathit{S}}^{♯}:T^{*}\mathit{S}\to T\mathit{S}:\mathrm{d}{f}_{\mid \mathit{x}}\mapsto {\mathit{g}}^{♯}(\mathrm{d}{f}_{\mid \mathit{x}})\in T_{\mathit{x}}\mathit{S}$$

(41)

such that

$${\mathit{g}}_{\mid \mathit{S}}(T{\iota }_S({\mathit{g}}_{T^{*}S}^{♯}(\mathrm{d}{f}_{\mid \mathit{x}})),{\mathit{v}}_x)=⟨\mathrm{d}{f}_{\mid \mathit{x}}\mid {\mathit{v}}_x⟩$$

(42)

for every $\mathrm{d}f\in C^{\infty }(\mathit{S})$ and every ${\mathit{v}}_{\mathit{x}}\in T_{\mathit{x}}{\mathrm{ℝ}}^n$ . The following diagram commutes:

$$\begin{array}{ccccc}\multicolumn{1}{c}{}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{T{\iota }_{\mathit{S}}}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{T\mathit{S}}& \multicolumn{1}{c}{\to }& \multicolumn{1}{c}{T_{\mathit{S}}{\mathrm{ℝ}}^d}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{{\mathit{g}}_{T\mathit{S}}^{♭}}& \multicolumn{1}{c}{\downarrow }& \multicolumn{1}{c}{}& \multicolumn{1}{c}{\downarrow }& \multicolumn{1}{c}{{\mathit{g}}_{\mid \mathit{S}}^{♭}}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{T^{*}\mathit{S}}& \multicolumn{1}{c}{\leftarrow }& \multicolumn{1}{c}{T_{\mathit{S}}^{*}{\mathrm{ℝ}}^d}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{T^{*}{\iota }_{\mathit{S}}}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}\end{array}.$$

(43)

The list above contains standard facts in geometry of ${\mathrm{ℝ}}^n$ . We prove one of the statements made here in order to prepare the ground for the case when $\mathit{S}\subseteq$ ${\mathrm{ℝ}}^d$ is not a vector subspace of ${\mathrm{ℝ}}^d$ but an arbitrary subset of ${\mathrm{ℝ}}^d$ . In order to verify that ${\mathit{g}}_{\mid \mathit{S}}^{♭}:T_{\mathit{S}}{\mathrm{ℝ}}^d\to T_{\mathit{S}}^{*}{\mathrm{ℝ}}^d$ restricted to $T\mathit{S}$ has range in $T^{*}\mathit{S}$ , observe that if ${\mathrm{v}}_{\mathit{x}}\in T_{\mathit{x}}\mathit{S}\subseteq T_S{\mathrm{ℝ}}^d$ , then

$$\begin{array}{cccc}\multicolumn{1}{c}{⟨{\mathit{g}}_{\mid \mathit{S}}^{♭}({\mathrm{v}}_{\mathit{x}})\mid {\mathrm{v}}_{\mathit{x}}^{\prime }⟩=0\forall {\mathrm{v}}_{\mathit{x}}^{\prime }\in T_{\mathit{x}}\mathit{S}}& \multicolumn{1}{c}{⟺}& \multicolumn{1}{c}{{\mathit{g}}_{\mid \mathit{S}}({\mathrm{v}}_{\mathit{x}},{\mathrm{v}}_{\mathit{x}}^{\prime })=0\forall {\mathrm{v}}_{\mathit{x}}^{\prime }\in T_{\mathit{x}}\mathit{S}}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{⟺}& \multicolumn{1}{c}{{\mathrm{v}}_{\mathit{x}}=0.}& \multicolumn{1}{c}{}\end{array}$$

Hence, for every ${\mathrm{v}}_x\in T\mathit{S}$ , there exists a unique covector $\mathrm{d}{f}_{\mathit{x}}\in T_{\mathit{x}}^{*}\mathit{S}$ such that ${\mathit{g}}_{\mid \mathit{S}}^{♭}({\mathrm{v}}_{\mathit{x}})=\mathrm{d}{f}_{\mathit{x}}$ .

4.3. Differential subspace $W\subseteq {\mathrm{ℝ}}^d$

Let W be a differential subspace of ${\mathrm{ℝ}}^d$ with the inclusion map ${\iota }_W:W\to {\mathrm{ℝ}}^d$ . The derived map $T{\iota }_W:\mathrm{TW}\to T{\mathrm{ℝ}}^d$ of the inclusion map is the inclusion of TW into $T{\mathrm{ℝ}}^d$ and ${\tau }_{{\mathrm{ℝ}}^d}\circ T{\iota }_W={\iota }_W\circ {\tau }_W$ . As in the preceding subsection, we denote by $T_W^{*}{\mathrm{ℝ}}^d$ the restriction of $T{\mathrm{ℝ}}^d$ to points in W, i.e., $T_W^{*}{\mathrm{ℝ}}^d={\pi }_{{\mathrm{ℝ}}^d}^{-1}(W)$ . The cotangent map of the inclusion map ${\iota }_W:W\to {\mathrm{ℝ}}^d$

$$T^{*}{\iota }_W:T_W^{*}{\mathrm{ℝ}}^d\to T^{*}W:\mathrm{d}{F}_{\mid \mathit{x}}\mapsto \mathrm{d}{({\iota }_W^{*}F)}_{\mid \mathit{x}},$$

(44)

maps $T_W^{*}{\mathrm{ℝ}}^d$ onto $T^{*}W$ . Hence, it defines in $T_W^{*}{\mathrm{ℝ}}^d$ an equivalence relation

$$\begin{array}{ccc}\multicolumn{1}{c}{\mathrm{d}{F}_{\mid \mathit{x}}\approx \mathrm{d}{F}_{\mid {\mathit{x}}^{\prime }}^{\prime }}& \multicolumn{1}{c}{⟺}& \multicolumn{1}{c}{T^{*}{\iota }_W(\mathrm{d}{F}_{\mid \mathit{x}})=T^{*}{\iota }_W(\mathrm{d}{F}_{\mid {\mathit{x}}^{\prime }}^{\prime })}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{⟺}& \multicolumn{1}{c}{\mathit{x}={\mathit{x}}^{\prime }\mathrm{and}⟨\mathrm{d}{F}_{\mid \mathit{x}}-\mathrm{d}{F}_{\mid \mathit{x}}^{\prime }\mid \mathrm{v}⟩=0\mathrm{for}\mathrm{all}\mathrm{v}\in T_{\mathit{x}}W.}& \multicolumn{1}{c}{}\end{array}$$

(45)

The kernel is this equivalence relation is the set of covectors in $T_W^{*}{\mathrm{ℝ}}^d$ , which are equivalent to zero. They form the annihilator ATW of TW in $T_W^{*}{\mathrm{ℝ}}^d$ ,

$$\mathrm{ATW}=\{\mathrm{d}{F}_{\mid \mathit{x}}\in T_W^{*}{\mathrm{ℝ}}^d\mid ⟨\mathrm{d}{F}_{\mid \mathit{x}}\mid \mathrm{v}⟩=0\forall \mathrm{v}\in \mathrm{TW}\}.$$

(46)

Hence, for every $\mathit{x}\in W$ , the quotient of $T_{\mathit{x}}^{*}{\mathrm{ℝ}}^d$ by $A{T}_{\mathit{x}}W$ is isomorphic to the cotangent space $T_{\mathit{x}}^{*}W$ ,

$$T_{\mathit{x}}^{*}W\equiv T_{\mathit{x}}^{*}{\mathrm{ℝ}}^d∕A{T}_{\mathit{x}}W.$$

(47)

Therefore, we may make an identification

$$T^{*}W=T^{*}{\mathrm{ℝ}}^d∕\mathrm{ATW}$$

(48)

and endow $T^{*}W$ with the quotient differential structure

$$C^{\infty }(T^{*}W)=C^{\infty }(T_W^{*}{\mathrm{ℝ}}^d∕\mathrm{ATW}).$$

(49)

Definition 4.1. The differential structure $C^{\infty }(T^{*}W)$ of $T^{*}W$ is the quotient differential structure corresponding to

$$T^{*}{\iota }_W:T_W^{*}{\mathrm{ℝ}}^n\to T^{*}W:\mathrm{d}{F}_{\mid \mathit{x}}\mapsto \mathrm{d}{({\iota }_W^{*}F)}_{\mid \mathit{x}}.$$

A function $h:T^{*}W\to \mathrm{ℝ}$ is in $C^{\infty }(T^{*}W)$ if and only if $h\circ T^{*}{\iota }_W:T_W^{*}{\mathrm{ℝ}}^d\to \mathrm{ℝ}$ is in $C^{\infty }(T_W^{*}{\mathrm{ℝ}}^d)$ .

In the quotient differential structure $C^{\infty }(T^{*}W)$ of $T^{*}W$ , the map $T^{*}{\iota }_W:T_W^{*}{\mathrm{ℝ}}^d\to T^{*}W$ is smooth. It is easy to see that if W is open in ${\mathrm{ℝ}}^d$ , then $T^{*}{\iota }_W:T_W^{*}{\mathrm{ℝ}}^d\to T^{*}W$ is a diffeomorphism. Our aim is to show that, with this definition of $C^{\infty }(T^{*}W)$ , we may adapt the approach from the preceding subsection to prove that $T^{*}W$ is diffeomorphic to TW.

Theorem 4.2. The restriction ${\mathit{g}}_{\mid \mathrm{TW}}^{♭}$ of

$${\mathit{g}}_{\mid W}^{♭}:T_W{\mathrm{ℝ}}^d\to T_W^{*}{\mathrm{ℝ}}^d:{\mathit{u}}_{\mathit{x}}\equiv {(u^1,\dots ,u^d)}_{\mathit{x}}\mapsto {\mathit{g}}_{\mid W}^{♭}({\mathit{u}}_{\mathit{x}})={\mathit{g}}^{♭}({\mathit{u}}_{\mathit{x}})$$

to $\mathrm{TW}\subseteq$ $T_W{\mathrm{ℝ}}^d$ gives a diffeomorphism

$${\mathit{g}}_{\mathrm{TW}}^{♭}={\mathit{g}}_{\mid \mathrm{TW}}^{♭}:\mathrm{TW}\to T^{*}W:{\mathrm{u}}_{\mathit{x}}\mapsto {\mathit{g}}_{\mid \mathrm{TW}}^{♭}({\mathrm{u}}_{\mathit{x}})$$

(50)

such that

$$⟨{\mathit{g}}_{\mathrm{TW}}^{♭}({\mathrm{u}}_{\mathit{x}})\mid {\mathrm{v}}_{\mathit{x}}⟩=\mathit{g}({\mathrm{u}}_{\mathit{x}},{\mathrm{v}}_{\mathit{x}})\forall {\mathrm{v}}_{\mathit{x}}\in T_{\mathit{x}}W\subseteq T_W{\mathrm{ℝ}}^d,$$

(51)

with inverse

$${\mathit{g}}_{T^{*}W}^{♯}:T^{*}W\to \mathrm{TW}:\mathrm{d}{f}_{\mid \mathit{x}}\mapsto {\mathit{g}}^{♯}(\mathrm{d}{f}_{\mid \mathit{x}})\in T_{\mathit{x}}W\subseteq T_W{\mathrm{ℝ}}^d$$

(52)

satisfying

$${\mathit{g}}_{\mid W}(T{\iota }_W({\mathit{g}}_{T^{*}W}^{♯}(\mathrm{d}{f}_{\mid \mathit{x}}),{\mathit{v}}_x)=⟨\mathrm{d}{f}_{\mid \mathit{x}}\mid {\mathit{v}}_x⟩$$

(53)

for every $f\in C^{\infty }(W)$ and ${\mathit{v}}_{\mathit{x}}\in T_{\mathit{x}}{\mathrm{ℝ}}^d$ .

Proof. In the preceding subsection, we showed that ${\mathit{g}}_{\mid S}^{♭}:T_S{\mathrm{ℝ}}^d\to T_S^{*}{\mathrm{ℝ}}^d$ restricted to TS has range in $T^{*}S$ . Since this argument is made separately at every point $\mathit{x}\in S$ , it is valid if we replace a vector subspace S of ${\mathrm{ℝ}}^d$ by an arbitrary subset W of ${\mathrm{ℝ}}^d$ . For $\mathit{x}\in W$ , if ${\mathrm{v}}_{\mathit{x}}\in T_{\mathit{x}}W\subseteq T_W{\mathrm{ℝ}}^d$ , then

$$\begin{array}{cccc}\multicolumn{1}{c}{⟨{\mathit{g}}_{\mid W}^{♭}({\mathrm{v}}_{\mathit{x}})\mid {\mathrm{v}}_{\mathit{x}}^{\prime }⟩=0\forall {\mathrm{v}}_{\mathit{x}}^{\prime }\in T_{\mathit{x}}W}& \multicolumn{1}{c}{⟺}& \multicolumn{1}{c}{{\mathit{g}}_{\mid W}({\mathrm{v}}_{\mathit{x}},{\mathrm{v}}_{\mathit{x}}^{\prime })=0\forall {\mathrm{v}}_{\mathit{x}}^{\prime }\in T_{\mathit{x}}W}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{⟺}& \multicolumn{1}{c}{{\mathrm{v}}_{\mathit{x}}=0.}& \multicolumn{1}{c}{}\end{array}$$

Hence, for every ${\mathrm{v}}_x\in \mathrm{TW}$ , there exists a unique covector $\mathrm{d}{f}_{\mathit{x}}\in T_{\mathit{x}}^{*}W$ such that ${\mathit{g}}_{\mid W}^{♭}({\mathrm{v}}_{\mathit{x}})=\mathrm{d}{f}_{\mathit{x}}$ . This justifies the statement in equation (50) that ${\mathit{g}}_{\mid \mathrm{TW}}^{♭}$ maps TW to $T^{*}W$ . For each $\mathit{x}\in W$ , the linear argument, used for $\mathit{x}\in S$ , implies that ${\mathit{g}}_{\mathrm{TW}\mid \mathit{x}}^{♭}:T_{\mathit{x}}W\to T_{\mathit{x}}^{*}W$ has the inverse ${\mathit{g}}_{T^{*}W\mid \mathit{x}}^{♯}:T_{\mathit{x}}^{*}W\to T_{\mathit{x}}W$ . Hence, ${\mathit{g}}_{\mathrm{TW}}^{♭}:T_{\mathit{x}}W\to T_{\mathit{x}}^{*}W$ and ${\mathit{g}}_{T^{*}W}^{♯}:T^{*}W\to \mathrm{TW}$ are linear isomorphisms, which are inverse of each other.

It remains to show that ${\mathit{g}}_{\mathrm{TW}}^{♭}:T_{\mathit{x}}W\to T_{\mathit{x}}^{*}W$ and ${\mathit{g}}_{T^{*}W}^{♯}:T^{*}W\to \mathrm{TW}$ are smooth. The following diagram commutes

$$\begin{array}{ccccc}\multicolumn{1}{c}{}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{{\mathit{g}}_{\mid W}^{♭}}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{T_W^{*}{\mathrm{ℝ}}^d}& \multicolumn{1}{c}{\leftarrow }& \multicolumn{1}{c}{T_W{\mathrm{ℝ}}^d}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{T^{*}{\iota }_W}& \multicolumn{1}{c}{\downarrow }& \multicolumn{1}{c}{}& \multicolumn{1}{c}{\uparrow }& \multicolumn{1}{c}{T{\iota }_W}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{T^{*}W}& \multicolumn{1}{c}{\leftarrow }& \multicolumn{1}{c}{\mathrm{TW}}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{{\mathit{g}}_{\mathrm{TW}}^{♭}}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}\end{array}.$$

(54)

This means that

$${\mathit{g}}_{\mathrm{TW}}^{♭}=T^{*}{\iota }_W\circ {\mathit{g}}_{\mid W}^{♭}\circ T{\iota }_W.$$

(55)

Smoothness of $T^{*}{\iota }_W$ is a direct consequence of Definition 4.1. ${\mathit{g}}_{\mid W}^{♭}$ is smooth because it is the restriction of the smooth map ${\mathit{g}}^{♭}:T{\mathrm{ℝ}}^d\to T^{*}{\mathrm{ℝ}}^d$ to the differential subspace $T_W{\mathrm{ℝ}}^d$ of $T{\mathrm{ℝ}}^d$ . Finally, $T{\iota }_W:\mathrm{TW}\to T{\mathrm{ℝ}}^d$ is smooth because ${\iota }_W:W\to {\mathrm{ℝ}}^d$ is smooth. Hence, ${\mathit{g}}_{\mathrm{TW}}^{♭}:\mathrm{TW}\to T^{*}W$ is a smooth map.

In order to prove that ${\mathit{g}}_{T^{*}W}^{♯}:T^{*}W\to \mathrm{TW}$ is smooth, we need to show that, for every $h\in C^{\infty }(\mathrm{TW})$ , the pull-back $h\circ {\mathit{g}}_{T^{*}W}^{♯}\in C^{\infty }(T^{*}W)$ . Since ${\mathit{g}}_{\mid T^{*}W}^{♯}={({\mathit{g}}_{\mathrm{TW}}^{♭})}^{-1}$ , the following diagram commutes:

$$\begin{array}{cccccc}\multicolumn{1}{c}{}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{{\mathit{g}}_{\mid W}^{♭}}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{T_W^{*}{\mathrm{ℝ}}^d}& \multicolumn{1}{c}{\leftarrow }& \multicolumn{1}{c}{T_W{\mathrm{ℝ}}^d}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{T^{*}{\iota }_W}& \multicolumn{1}{c}{\downarrow }& \multicolumn{1}{c}{}& \multicolumn{1}{c}{\uparrow }& \multicolumn{1}{c}{T{\iota }_W}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{T^{*}W}& \multicolumn{1}{c}{\to }& \multicolumn{1}{c}{\mathrm{TW}}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{{\mathit{g}}_{T^{*}W}^{♯}}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{\searrow }& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{h}& \multicolumn{1}{c}{\mathrm{ℝ}}\end{array}.$$

(56)

Following the arrows, we see that

$$T{\iota }_W\circ {\mathit{g}}_{T^{*}W}^{♯}\circ T^{*}{\iota }_W\circ {\mathit{g}}_{\mid W}^{♭}={\mathrm{id}}_{T_W{\mathrm{ℝ}}^d}$$

(57)

By the definition of $C^{\infty }(T^{*}W)$ , $h\circ {\mathit{g}}_{T^{*}W}^{♯}$ is in $C^{\infty }(T^{*}W)$ if its pull-back $h\circ {\mathit{g}}_{T^{*}W}^{♯}\circ T^{*}{\iota }_W:T_W^{*}{\mathrm{ℝ}}^d\to \mathrm{ℝ}$ is in $C^{\infty }(T_W^{*}{\mathrm{ℝ}}^d)$ . Since ${\mathit{g}}_{\mid W}^{♭}:T_W{\mathrm{ℝ}}^d\to T_W^{*}{\mathrm{ℝ}}^d$ is a diffeomorphism, it follows that

$$(h\circ {\mathit{g}}_{T^{*}W}^{♯})\in C^{\infty }(T^{*}W)⟺h\circ {\mathit{g}}_{T^{*}W}^{♯}\circ T^{*}{\iota }_W\circ {\mathit{g}}_{\mid W}^{♭}\in C^{\infty }(T_W{\mathrm{ℝ}}^d).$$

(58)

TW is a differential subspace of $T_W{\mathrm{ℝ}}^d$ with the inclusion map $T{\iota }_W:\mathrm{TW}\to T{\mathrm{ℝ}}^d$ restricted in co-domain to $T_W{\mathrm{ℝ}}^d$ . Hence, $C^{\infty }(\mathrm{TW})$ is generated by the restrictions to TW of functions in $C^{\infty }(T_W{\mathrm{ℝ}}^d)$ . Suppose first that $h=H_{\mid \mathrm{TW}}=H\circ T{\iota }_W$ for some $H\in C^{\infty }(T_W{\mathrm{ℝ}}^d)$ . Therefore,

$$h\circ {\mathit{g}}_{T^{*}W}^{♯}=H_{\mid \mathrm{TW}}\circ {\mathit{g}}_{T^{*}W}^{♯}=H\circ T{\iota }_W\circ {\mathit{g}}_{T^{*}W}^{♯},$$

(59)

and

$$\begin{array}{ccc}\multicolumn{1}{c}{}& \multicolumn{1}{c}{(h\circ {\mathit{g}}_{T^{*}W}^{♯})=}& \multicolumn{1}{c}{H\circ T{\iota }_W\circ {\mathit{g}}_{T^{*}W}^{♯}\in C^{\infty }(T^{*}W)}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{⟺}& \multicolumn{1}{c}{H\circ T{\iota }_W\circ {\mathit{g}}_{T^{*}W}^{♯}\circ \mathrm{Corollary}{T}^{*}{\iota }_W\circ {\mathit{g}}_{\mid W}^{♭}=H\in C^{\infty }(T_W{\mathrm{ℝ}}^d),}& \multicolumn{1}{c}{}\end{array}$$

(60)

where we have used equation (57). The assumption that $H\in C^{\infty }(T_W{\mathrm{ℝ}}^d)$ implies that $H\circ T{\iota }_W\circ {\mathit{g}}_{T^{*}W}^{♯}\in C^{\infty }(T^{*}W)$ .

Every $h\in C^{\infty }(\mathrm{TW})$ is locally the restriction to TW of a function in $C^{\infty }(T_W{\mathrm{ℝ}}^d)$ . Therefore, for each ${\mathit{v}}_{\mathit{x}}\in \mathrm{TW}$ , there exists a neighbourhood $U^{\prime }$ of ${\mathit{v}}_{\mathit{x}}$ in TW and a function $H^{\prime }\in$ $C^{\infty }(T_W{\mathrm{ℝ}}^d)$ such that $h_{\mid U^{\prime }}=H_{\mid U^{\prime }}^{\prime }$ . Repeating the argument above for $h_{\mid U^{\prime }}$ , we conclude that $h_{\mid U^{\prime }}\circ {\mathit{g}}_{T^{*}W}^{♯}$ is in $C^{\infty }({\mathit{g}}_{T^{*}W}^{♯}(U^{\prime }))$ . Since it holds for every ${\mathit{v}}_{\mathit{x}}$ in TW, and smoothness is a local property, it follows that $h\circ {\mathit{g}}_{T^{*}W}^{♯}\in C^{\infty }(T^{*}W)$ for every $h\in C^{\infty }(\mathrm{TW})$ . Hence, ${\mathit{g}}_{T^{*}W}^{♯}:T^{*}W\to \mathrm{TW}$ is smooth, which completes the proof. ■

Corollary 4.3. Since TW is a differential subspace of $T{\mathrm{ℝ}}^d$ , which is diffeomorphic to ${\mathrm{ℝ}}^{2d}$ , the proof that ${\mathit{g}}_{T^{*}W}^{♯}:T^{*}W\to \mathrm{TW}$ is a diffeomorphism ensures that $T^{*}W$ is diffeomorphic to a differential subspace of ${\mathrm{ℝ}}^{2d}$ . Hence, all properties of locally Euclidean differential spaces, discussed at the beginning of the paper, apply to cotangent bundles of locally Euclidean differential spaces. ⁶

Theorem 4.4. For every $f\in C^{\infty }(W)$ , where W is a differential subspace of ${\mathrm{ℝ}}^d$ , the differential $\mathrm{d}f:W\to T^{*}W:x\mapsto \mathrm{d}{f}_{\mid x}$ is smooth.

Proof. In order to prove that $\mathrm{d}f:W\to T^{*}W$ is smooth, we need to show that, for every $h\in C^{\infty }(T^{*}W)$ , the pull-back $h\circ \mathrm{d}f:W\to \mathrm{ℝ}$ is in $C^{\infty }(W)$ .

Consider first the case when W is closed in ${\mathrm{ℝ}}^d$ . By Definition 4.1, $h\in C_{\mathrm{\setminus }}^{\infty }(T^{*}W)$ implies that $h\circ T^{*}{\iota }_W:T_W^{*}{\mathrm{ℝ}}^d\to \mathrm{ℝ}$ is in $C^{\infty }(T_W^{*}{\mathrm{ℝ}}^d)$ . But, $T_W^{*}{\mathrm{ℝ}}^d$ is also closed in $T^{*}{\mathrm{ℝ}}^d$ and there exists a function $H\in C^{\infty }(T^{*}{\mathrm{ℝ}}^d)$ such that $h\circ T^{*}{\iota }_W=H_{\mid T_W^{*}{\mathrm{ℝ}}^d}$ . Furthermore, $f\in C^{\infty }(W)$ extends to a function $F\in C^{\infty }({\mathrm{ℝ}}^d)$ . In other words, $f=F_{\mid W}$ . Moreover, $\mathrm{d}f=\mathrm{d}F\mid \mathrm{TW}$ . But $\mathrm{d}F:{\mathrm{ℝ}}^d\to T^{*}{\mathrm{ℝ}}^d$ is smooth. Hence, the restriction ${(\mathrm{d}F)}_{\mid W}:W\to T_W^{*}{\mathrm{ℝ}}^d$ is smooth. Since $H\in C^{\infty }(T^{*}{\mathrm{ℝ}}^d)$ , it follows that $H\circ \mathrm{d}F\in C^{\infty }({\mathrm{ℝ}}^d)$ . Therefore, the restriction ${(H\circ \mathrm{d}F)}_{\mid W}\in C^{\infty }(W)$ . But,

$${(H\circ \mathrm{d}F)}_{\mid W}=H_{\mid T_W^{*}{\mathrm{ℝ}}^d}\circ {(\mathrm{d}F)}_{\mid W}=h\circ T^{*}{\iota }_W\circ {(\mathrm{d}F)}_{\mid W}.$$

By equation (44), for every $\mathit{x}\in W$ ,

$$h\circ T^{*}{\iota }_W\circ (\mathrm{d}{F}_{\mid \mathit{x}})=h\circ \mathrm{d}{({\iota }_W^{*}F)}_{\mid \mathit{x}}=h\circ \mathrm{d}{(F_{\mid W})}_{\mid \mathit{x}}=h(\mathrm{d}{f}_{\mid \mathit{x}}).$$

Hence,

$$h\circ \mathrm{d}f=h\circ T^{*}{\iota }_W\circ {(\mathrm{d}F)}_{\mid W}={(H\circ \mathrm{d}F)}_{\mid W}\in C^{\infty }(W).$$

This holds for every $f\in C^{\infty }(W)$ and every $h\in C^{\infty }(T^{*}W)$ . Therefore, the map $\mathrm{d}f:W\to T^{*}W:x\mapsto \mathrm{d}{f}_{\mid x}$ is smooth in the differential structure $C^{\infty }(T^{*}W)$ .

If W is not closed in ${\mathrm{ℝ}}^n$ , then the argument above shows that $h\circ \mathrm{d}f$ is locally smooth. That is, for every $\mathit{x}\in W$ , there exists an open neighbourhood $U_{\mathit{x}}$ of $\mathit{x}$ in ${\mathrm{ℝ}}^n$ , $H_{\mathit{x}}\in C^{\infty }(T^{*}{\mathrm{ℝ}}^n)$ and $F_{\mathit{x}}\in C^{\infty }({\mathrm{ℝ}}^n)$ such that

$${(h\circ \mathrm{d}f)}_{\mid W\cap U_{\mathit{x}}}={(H_{\mathit{x}}\circ \mathrm{d}{F}_{\mathit{x}})}_{\mid W\cap U_{\mathit{x}}}\mathrm{and}{H}_{\mathit{x}}\circ \mathrm{d}{F}_{\mathit{x}}\in C^{\infty }({\mathrm{ℝ}}^n).$$

Since the differential structure $C^{\infty }(W)$ is generated by the restrictions to W of functions in $C^{\infty }({\mathrm{ℝ}}^n)$ , it follows that $h\circ \mathrm{d}f\in C^{\infty }(W)$ . This holds for every $f\in C^{\infty }(W)$ and every $h\in C^{\infty }(T^{*}W)$ . Therefore, the map $\mathrm{d}f:W\to T^{*}W:x\mapsto \mathrm{d}{f}_{\mid x}$ is smooth in the quotient differential structure $C^{\infty }(T^{*}W)$ . ■

Definition 4.5. A section of the cotangent bundle $T^{*}W$ is a smooth map $\vartheta :W\to T^{*}W$ such that ${\pi }_W\circ \vartheta ={\mathrm{id}}_W$ , where ${\pi }_W:T^{*}W\to W$ is the cotangent bundle projection and ${\mathrm{id}}_W:W\to W:\mathit{x}\mapsto \mathit{x}$ is the identity map. We denote by $\mathrm{Sec}(T^{*}W)$ the space of sections of $T^{*}W$ .

Claim 4.6. The space $\mathrm{Sec}(T^{*}W)$ of sections of $T^{*}W$ is closed under the addition of sections and the multiplications of sections by smooth functions.

Proof. The musical morphisms ${\mathit{g}}^{♯}:T^{*}{\mathrm{ℝ}}^d\to T{\mathrm{ℝ}}^d$ and ${\mathit{g}}^{♭}:T{\mathrm{ℝ}}^d\to T^{*}{\mathrm{ℝ}}^d$ commute with the operations of the addition of sections and the multiplication of sections by smooth functions. Moreover, they commute with the tangent and the cotangent bundle projections; i.e., the following diagrams commute:

$$\begin{array}{ccccc}\multicolumn{1}{c}{}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{{\mathit{g}}^{♯}}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{T^{*}{\mathrm{ℝ}}^d}& \multicolumn{1}{c}{\to }& \multicolumn{1}{c}{T{\mathrm{ℝ}}^d}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{{\pi }_{{\mathrm{ℝ}}^d}}& \multicolumn{1}{c}{\downarrow }& \multicolumn{1}{c}{}& \multicolumn{1}{c}{\downarrow }& \multicolumn{1}{c}{{\tau }_{{\mathrm{ℝ}}^d}}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{{\mathrm{ℝ}}^d}& \multicolumn{1}{c}{=}& \multicolumn{1}{c}{{\mathrm{ℝ}}^d}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}\end{array}\mathrm{and}\begin{array}{ccccc}\multicolumn{1}{c}{}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{{\mathit{g}}^{♭}}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{T{\mathrm{ℝ}}^d}& \multicolumn{1}{c}{\to }& \multicolumn{1}{c}{T^{*}{\mathrm{ℝ}}^d}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{{\tau }_{{\mathrm{ℝ}}^d}}& \multicolumn{1}{c}{\downarrow }& \multicolumn{1}{c}{}& \multicolumn{1}{c}{\downarrow }& \multicolumn{1}{c}{{\pi }_{{\mathrm{ℝ}}^d}}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{{\mathrm{ℝ}}^d}& \multicolumn{1}{c}{=}& \multicolumn{1}{c}{{\mathrm{ℝ}}^d}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}\end{array}.$$

The restriction ${\mathit{g}}_{\mid \mathrm{TM}}^{♭}$ of ${\mathit{g}}^{♭}$ to TM gives a diffeomorphism ${\mathit{g}}_{\mid \mathrm{TM}}^{♭}:\mathrm{TM}$ → $T^{*}M$ with inverse ${\mathit{g}}_{T^{*}M}^{♯}:T^{*}M$ → TM, which intertwine the operations of the addition of sections and the multiplication of sections by smooth functions. ■

Proposition 4.7. Every $X\in \mathrm{Der}{C}^{\infty }(W)$ corresponds to a smooth function, also denoted by X,

$$X:T^{*}W\to \mathrm{ℝ}:\mathrm{d}{f}_{\mid \mathit{x}}\mapsto X(\mathrm{d}{f}_{\mid \mathit{x}})=⟨\mathrm{d}{f}_{\mid \mathit{x}}\mid X(\mathit{x})⟩=(\mathrm{Xf})(\mathit{x}).$$

Proof. For every $\mathrm{d}{F}_{\mid \mathit{x}}\in T_W^{*}{\mathrm{ℝ}}^d$ ,

$$\begin{array}{cccc}\multicolumn{1}{c}{X\circ T^{*}{\iota }_W(\mathrm{d}{F}_{\mid \mathit{x}})}& \multicolumn{1}{c}{=}& \multicolumn{1}{c}{⟨T^{*}{\iota }_W(\mathrm{d}{F}_{\mid \mathit{x}})\mid X(\mathit{x})⟩=⟨\mathrm{d}{F}_{\mid \mathit{x}}\mid T{\iota }_W(X(\mathit{x}))⟩}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{=}& \multicolumn{1}{c}{⟨\mathrm{d}{F}_{\mid \mathit{x}}\mid (T{\iota }_W\circ X)(\mathit{x})⟩.}& \multicolumn{1}{c}{}\end{array}$$

Since $W\subseteq {\mathrm{ℝ}}^d$ , every $X\in \mathrm{Der}{\mathcal{C}}^{\infty }(W)$ locally extends to a vector field on ${\mathrm{ℝ}}^d$ . ⁷ That is, for every ${\mathit{x}}_0\in W$ , there exists an open neighbourhood U of ${\mathit{x}}_0$ in ${\mathrm{ℝ}}^d$ and a vector field Y on ${\mathrm{ℝ}}^n$ such that ${(T{\iota }_W\circ X)}_{\mid U\cap W}=Y_{\mid U\cap W}$ . Hence,

$${(X\circ T^{*}{\iota }_W)}_{\mid {\pi }_{{\mathrm{ℝ}}^n}^{-1}(U\cap V)}=Y_{\mid {\pi }_{{\mathrm{ℝ}}^n}^{-1}(U\cap V)}.$$

(61)

Since $Y\in C^{\infty }(T^{*}{\mathrm{ℝ}}^d)$ and equation (61) holds in neighbourhood of every ${\mathit{x}}_0\in W$ , it follows that $X\circ T^{*}{\iota }_W\in C^{\infty }(T_W^{*}{\mathrm{ℝ}}^d)$ , which implies that $X\in C^{\infty }(T^{*}W)$ . ■

Remark 4.8. In section 3, we associated with a one-to-one smooth map $\phi :S\to R$ of differential spaces its cotangent map $T^{*}\phi :T_{\phi (S)}^{*}R\to T^{*}S$ , see Definition 3.4 and Lemma 3.5. We used it in Definition 4.1, in the case when ${\iota }_W:W\to {\mathrm{ℝ}}^d$ is the inclusion map, to define $C^{\infty }(T^{*}W)$ as the space of functions $h:T^{*}W\to \mathrm{ℝ}$ such that $h\circ T^{*}{\iota }_W:T_W^{*}{\mathrm{ℝ}}^d\to \mathrm{ℝ}$ is in $C^{\infty }(T_W^{*}{\mathrm{ℝ}}^d)$ .

Suppose that $\phi :V\to W$ is a diffeomorphism of closed differential subspaces of ${\mathrm{ℝ}}^d$ , with inclusion maps ${\iota }_V:V\to {\mathrm{ℝ}}^d$ and ${\iota }_W:W\to {\mathrm{ℝ}}^d$ . It follows from Whitney extension theorem [17] that $\phi :V\to W$ lifts to a diffeomorphism $\mathrm{\Phi }:T_W^{*}{\mathrm{ℝ}}^d$ $\to T_V^{*}{\mathrm{ℝ}}^d$ and the following diagram commutes for every $f\in C^{\infty }(T^{*}V)$ ,

$$\begin{array}{ccccc}\multicolumn{1}{c}{}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{\mathrm{ℝ}}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{\uparrow f}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{T_V^{*}{\mathrm{ℝ}}^d}& \multicolumn{1}{c}{\underset{T^{*}{\iota }_V}{\mathrm{\to }}}& \multicolumn{1}{c}{T^{*}V}& \multicolumn{1}{c}{\underset{{\pi }_V}{\mathrm{\to }}}& \multicolumn{1}{c}{V}\\ \multicolumn{1}{c}{\uparrow \mathrm{\Phi }}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{\uparrow T^{*}\phi }& \multicolumn{1}{c}{}& \multicolumn{1}{c}{\downarrow \phi }\\ \multicolumn{1}{c}{T_W^{*}{\mathrm{ℝ}}^d}& \multicolumn{1}{c}{\underset{T^{*}{\iota }_W}{\mathrm{\to }}}& \multicolumn{1}{c}{T^{*}W}& \multicolumn{1}{c}{\underset{{\pi }_W}{\mathrm{\to }}}& \multicolumn{1}{c}{W}\end{array}.$$

Hence, $f\circ T^{*}\phi \circ T^{*}{\iota }_W=f\circ T^{*}{\iota }_W\circ \mathrm{\Phi }$ , which implies that $f\circ T^{*}\phi \circ T^{*}{\iota }_W:T_W^{*}{\mathrm{ℝ}}^d\to \mathrm{ℝ}$ . By Definition 4.1, $f\circ T^{*}\phi \in C^{\infty }(T^{*}W)$ . Since this holds for every $f\in C^{\infty }(T^{*}V)$ , it follows that $T^{*}\phi :T^{*}W\to T^{*}V$ is smooth.

4.4. Cotangent bundle of a locally Euclidean differential space

Let $\mathfrak{A}(S)=\{\alpha :V_{\alpha }\to W_{\alpha }\subseteq {\mathrm{ℝ}}^{d_{\alpha }}\}$ be an atlas for a locally Euclidean differential space S. Here, $V_{\alpha }$ is an open differential subspace of S, $W_{\alpha }$ is a differential subspace of ${\mathrm{ℝ}}^{d_{\alpha }}$ and $\alpha :V_{\alpha }\to W_{\alpha }$ is a diffeomorphism. Let ${\pi }_S:T^{*}S\to S:\mathrm{d}{f}_{\mid x}\mapsto x$ , ${\pi }_{V_{\alpha }}:T^{*}{V}_{\alpha }\to V_{\alpha }$ , ${\pi }_{W_{\alpha }}:T^{*}{W}_{\alpha }\to W_{\alpha }$ , and ${\pi }_{{\mathrm{ℝ}}^{d_{\alpha }}}:T^{*}{\mathrm{ℝ}}^{d_{\alpha }}\to {\mathrm{ℝ}}^{d_{\alpha }}$ denote the cotangent bundle projections of the cotangent bundles of the spaces occurring in the atlas. Consider the following diagram:

$$\begin{array}{ccccccccc}\multicolumn{1}{c}{}& \multicolumn{1}{c}{{\iota }_{T_{V_{\alpha }}^{*}S}}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{T^{*}{\iota }_{V_{\alpha }}}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{T^{*}\alpha }& \multicolumn{1}{c}{}& \multicolumn{1}{c}{T^{*}{\iota }_{W_{\alpha }}}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{T^{*}S}& \multicolumn{1}{c}{\leftarrow }& \multicolumn{1}{c}{T_{V_{\alpha }}^{*}S}& \multicolumn{1}{c}{\to }& \multicolumn{1}{c}{T^{*}{V}_{\alpha }}& \multicolumn{1}{c}{\leftarrow }& \multicolumn{1}{c}{T^{*}{W}_{\alpha }}& \multicolumn{1}{c}{\leftarrow }& \multicolumn{1}{c}{T_{W_{\alpha }}^{*}{\mathrm{ℝ}}^{d_{\alpha }}}\\ \multicolumn{1}{c}{{\pi }_S\downarrow }& \multicolumn{1}{c}{}& \multicolumn{1}{c}{\downarrow {\widehat{\pi }}_{V_{\alpha }}}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{{\pi }_{V_{\alpha }}\downarrow }& \multicolumn{1}{c}{}& \multicolumn{1}{c}{\downarrow {\pi }_{W_{\alpha }}}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{{\widehat{\pi }}_{W_{\alpha }}\downarrow }\\ \multicolumn{1}{c}{S}& \multicolumn{1}{c}{\leftarrow }& \multicolumn{1}{c}{V_{\alpha }}& \multicolumn{1}{c}{=}& \multicolumn{1}{c}{V_{\alpha }}& \multicolumn{1}{c}{\to }& \multicolumn{1}{c}{W_{\alpha }}& \multicolumn{1}{c}{=}& \multicolumn{1}{c}{W_{\alpha }}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{{\iota }_{V_{\alpha }}}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{\alpha }& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}\end{array}.$$

The bottom row corresponds to a chart α on $S$ expanded by the addition of the inclusion map ${\iota }_{V_{\alpha }}:V_{\alpha }\hookrightarrow S$ . The inclusion map ${\iota }_{W_{\alpha }}:W_{\alpha }\hookrightarrow {\mathrm{ℝ}}^{d_{\alpha }}$ is omitted due to lack of space. All the maps corresponding to the horizontal arrows in the bottom row are smooth. The vertical arrows correspond to appropriate cotangent bundle projections. The top row consists of inclusion maps, the cotangent lifts of inclusion maps, and the cotangent lift of a diffeomorphism. We follow the diagram above from right to left.

Definition 4.9. For every $\alpha \in \mathfrak{A}$ ,

• $T^{*}{W}_{\alpha }$ has the quotient differential structure $C^{\infty }(T^{*}{W}_{\alpha })$ corresponding to the map $T^{*}{\iota }_{W_{\alpha }}:T_{\mathrm{W\alpha }}^{*}{\mathrm{ℝ}}^{d_{\alpha }}\to T^{*}{W}_{\alpha }$ .

• $T^{*}{V}_{\alpha }$ has a differential structure $C^{\infty }(T^{*}{V}_{\alpha })$ induced by the bijection $T^{*}\alpha :T^{*}{W}_{\alpha }\to T^{*}{V}_{\alpha }$ .

A function $f:T^{*}{V}_{\alpha }\to \mathrm{ℝ}$ is in $C^{\infty }(T^{*}{V}_{\alpha })$ if $f\circ T^{*}\alpha \in T^{*}{W}_{\alpha }$ . Since $\alpha :V_{\alpha }\to W_{\alpha }$ is a diffeomorphism, it follows that $T^{*}\alpha :T^{*}{W}_{\alpha }\to T^{*}{V}_{\alpha }$ is a diffeomorphism. In particular, ${\pi }_{V_{\alpha }}:T^{*}{V}_{\alpha }\to V_{\alpha }$ is smooth.

By definition of the atlas, $V_{\alpha }$ is an open differential subspace of S. Therefore, Lemma 3.5 ensures that $T^{*}{V}_{\alpha }=T_{V_{\alpha }}^{*}S$ and $T^{*}{\iota }_{V_{\alpha }}={\mathrm{id}}_{T^{*}{V}_{\alpha }}$ . Hence, ${\widehat{\pi }}_{V_{\alpha }}={\pi }_{V_{\alpha }}$ and ${\iota }_{T_{V_{\alpha }}^{*}S}={\iota }_{T^{*}{V}_{\alpha }}$ . This allows us to rewrite the last square from the right of the above diagram in the form:

$$\begin{array}{ccc}\multicolumn{1}{c}{}& \multicolumn{1}{c}{{\iota }_{T^{*}{V}_{\alpha }}}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{T^{*}S}& \multicolumn{1}{c}{\leftarrow }& \multicolumn{1}{c}{T^{*}{V}_{\alpha }}\\ \multicolumn{1}{c}{{\pi }_S\downarrow }& \multicolumn{1}{c}{}& \multicolumn{1}{c}{\downarrow {\pi }_{V_{\alpha }}}\\ \multicolumn{1}{c}{S}& \multicolumn{1}{c}{\leftarrow }& \multicolumn{1}{c}{V_{\alpha }}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{{\iota }_{V_{\alpha }}}& \multicolumn{1}{c}{}\end{array}.$$

(62)

By assumption, S has the differential structure $C^{\infty }(S)$ of a locally Euclidean differential space, and $V_{\alpha }$ is an open differential subspace of S. By the arguments above, the differential structure $C^{\infty }(T^{*}{V}_{\alpha })$ of $T^{*}{V}_{\alpha }$ is induced by the diffeomorphism $T^{*}\alpha :T^{*}{W}_{\alpha }\to T^{*}{V}_{\alpha }$ . Commutativity of diagram (62), for every $\alpha \in \mathfrak{A}(S)$ , determines the differential structure of $T^{*}S$ .

Definition 4.10. A function $h:T^{*}S\to \mathrm{ℝ}$ is in $C^{\infty }(T^{*}S)$ if, for every α $\in \mathfrak{A}(S)$ , the restriction $h_{\mid T^{*}{V}_{\alpha }}\in C^{\infty }(T^{*}{V}_{\alpha })$ . Thus,

$$C^{\infty }(T^{*}S)=\{h:T^{*}S\to \mathrm{ℝ}\mid h_{\mid T^{*}{V}_{\alpha }}\in C^{\infty }(T^{*}{V}_{\alpha })\forall \alpha \in \mathfrak{A}(S)\}.$$

(63)

Proposition 4.11. In the differential structure $C^{\infty }(T^{*}S)$ given by equation (63), the commutativity of diagram (62) implies that ${\pi }_S:T^{*}S\to S$ is smooth so that, for each $\alpha \in \mathfrak{A}(S)$ , $T^{*}{V}_{\alpha }=T_{V_{\alpha }}^{*}S={\pi }_S^{-1}(V_{\alpha })$ is open in $T^{*}S$ .

Proof. For every $f\in C^{\infty }(S)$ and every $\alpha \in \mathfrak{A}(S)$ ,

$${({\pi }_S^{*}f)}_{\mid T^{*}{V}_{\alpha }}=f\circ {\pi }_S\circ {\iota }_{T^{*}{V}_{\alpha }}=f\circ {\iota }_{V_{\alpha }}\circ {\pi }_{V_{\alpha }}\in C^{\infty }(T^{*}{V}_{\alpha }).$$

Therefore, ${\pi }_S:T^{*}S\to S$ is smooth. It follows from Lemma 3.5 that, for each $\alpha \in \mathfrak{A}(S)$ , $T^{*}{V}_{\alpha }=T_{V_{\alpha }}^{*}S={\pi }_S^{-1}(V_{\alpha })$ is open in $T^{*}S$ . ■

Proposition 4.12. $C^{\infty }(T^{*}S)$ , given by equation (63), satisfies all conditions of Definition 2.1.

Proof. Since $C^{\infty }(T^{*}{V}_{\alpha })$ determines a sub-basis for the topology of $V_{\alpha }$ for every $\alpha \in \mathfrak{A}(S)$ , and $\{V_{\alpha }\mid \alpha \in$ $\mathfrak{A}(S)\}$ is an open cover of S, it follows that $C^{\infty }(T^{*}S)$ determines a sub-basis for the topology of S, as required by Definition 2.1.

Let $h_1,\dots ,h_n\in C^{\infty }(T^{*}S)$ and $F\in C^{\infty }({\mathrm{ℝ}}^n)$ . For every $\alpha \in \mathfrak{A}(S)$ , the restriction $F{(h_1,\dots ,h_n)}_{\mid T^{*}{V}_{\alpha }}=F(h_{1\mid T^{*}{V}_{\alpha }},\dots ,h_{n\mid T^{*}{V}_{\alpha }})\in C^{\infty }(T^{*}{V}_{\alpha })$ . Hence, $F(h_1,\dots ,h_n)\in C^{\infty }(T^{*}S)$ .

Suppose that $h:T^{*}S\to \mathrm{ℝ}$ is a function on $T^{*}S$ such that, for each $p\in T^{*}S$ , there is an open neighbourhood $U_p$ of p in $T^{*}S$ and a function $h^{\prime }\in C^{\infty }(T^{*}S)$ satisfying $h_{\mid U_p}=h_{\mid U_p}^{\prime }$ . Then, for every $\alpha \in \mathfrak{A}(S)$ , and every $p\in T^{*}{V}_{\alpha }$ ,

$${(h_{\mid T^{*}{V}_{\alpha }})}_{\mid U_p\cap T^{*}{V}_{\alpha }}=h_{\mid T^{*}{V}_{\alpha }\cap U_p}=h_{\mid T^{*}{V}_{\alpha }\cap \mathrm{Up}}^{\prime }={(h_{\mid T^{*}{V}_{\alpha }}^{\prime })}_{\mid U_p\cap T^{*}{V}_{\alpha }}.$$

(64)

Since $h^{\prime }\in C^{\infty }(T^{*}S)$ , Definition 4.6 implies that $h_{\mid T^{*}{V}_{\alpha }}^{\prime }\in C^{\infty }(T^{*}{V}_{\alpha })$ . By Definition 2.1(3), equation (64) ensures that $h_{\mid T^{*}{V}_{\alpha }}\in C^{\infty }(T^{*}{V}_{\alpha })$ . Hence, $h\in C^{\infty }(T^{*}S)$ , as required. ■

Theorem 4.13. The cotangent bundle $T^{*}S$ of a locally Euclidean differential space S, endowed with the differential structure $C^{\infty }(T^{*}S)$ described in Definitions 4.9 and 4.10, is locally Euclidean.

Proof. By Definition 4.9 (i), for every $\alpha \in \mathfrak{A}$ , $T^{*}{W}_{\alpha }$ has the quotient differential structure $C^{\infty }(T^{*}{W}_{\alpha })$ corresponding to the map $T^{*}{\iota }_{W_{\alpha }}:T_{\mathrm{W\alpha }}^{*}{\mathrm{ℝ}}^{d_{\alpha }}\to T^{*}{W}_{\alpha }$ . Theorem 4.2 ensures that $T^{*}{W}_{\alpha }$ is diffeomorphic to $T{W}_{\alpha }$ , which is a differential subspace of ${\mathrm{ℝ}}^{2d_{\alpha }}$ . Moreover, by Definition 4.9 (ii), $T^{*}{V}_{\alpha }$ is diffeomorphic to $T^{*}{W}_{\alpha }$ . Proposition 4.11 ensures that $T^{*}{V}_{\alpha }$ is an open subset of $T^{*}S$ . Since $T^{*}S={\mathrm{\cup }}_{\alpha \in \mathfrak{A}}{T}^{*}{V}_{\alpha }$ , it follows that $T^{*}S$ is locally Euclidean. ■

5. Sections and forms

In the following, we assume that every cotangent bundle $T^{*}S$ of a locally Euclidean differential space S, considered in this section, is endowed with the differential structure $C^{\infty }(T^{*}S)$ described in the preceding section.

5.1. Sections

Definition 5.1. A section of the cotangent bundle $T^{*}S$ of S is a smooth map $\vartheta :S\to T^{*}S$ , such that ${\pi }_S\circ \vartheta ={\mathrm{id}}_S$ , where ${\pi }_S:T^{*}S\to S$ is the cotangent bundle projection and ${\mathrm{id}}_S:S\to S$ is the identity mapping. A local section of $T^{*}S$ is a map $\vartheta :V\to T_V^{*}S\equiv T^{*}V$ , where V is an open differential subspace of S, such that ${\pi }_V\circ \vartheta ={\mathrm{id}}_V$ .

Theorem 5.2. Let $T^{*}S$ be the cotangent bundle of a locally Euclidean differential space S. For every $f\in C^{\infty }(S)$ , the differential

$$\mathrm{d}f:S\to T^{*}S:x\mapsto \mathrm{d}{f}_{\mid x}$$

is a smooth section of $T^{*}S$ .

Proof. Let $\mathfrak{A}(S)=\{\alpha :V_{\alpha }\to W_{\alpha }\subseteq {\mathrm{ℝ}}^{d_{\alpha }}\}$ be an atlas for S. For every $\alpha \in \mathfrak{A}(S)$ , by Proposition 4.7, $T_{V_{\alpha }}^{*}S=T^{*}{V}_{\alpha }$ so that $\mathrm{d}{f}_{\mid V_{\alpha }}:V_{\alpha }\to T^{*}{V}_{\alpha }$ . Since $\alpha :V_{\alpha }\to W_{\alpha }$ is a diffeomorphism, its tangent map $T^{*}\alpha :T^{*}{W}_{\alpha }\to T^{*}{V}_{\alpha }$ is a diffeomorphism and the following diagram commutes:

$$\begin{array}{ccccc}\multicolumn{1}{c}{}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{T^{*}{\alpha }^{-1}}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{T^{*}{V}_{\alpha }}& \multicolumn{1}{c}{\to }& \multicolumn{1}{c}{T^{*}{W}_{\alpha }}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{\mathrm{d}{f}_{\mid V_{\alpha }}}& \multicolumn{1}{c}{\uparrow }& \multicolumn{1}{c}{}& \multicolumn{1}{c}{\downarrow }& \multicolumn{1}{c}{\mathrm{d}(f\circ {\alpha }^{-1})}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{V_{\alpha }}& \multicolumn{1}{c}{\leftarrow }& \multicolumn{1}{c}{W_{\alpha }}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{{\alpha }^{-1}}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}\end{array}.$$

But, $f\circ {\alpha }^{-1}$ is in $C^{\infty }(W_{\alpha })$ and $\mathrm{d}(f\circ {\alpha }^{-1}):W_{\alpha }\to T^{*}{W}_{\alpha }$ is smooth by Proposition 4.3. Hence,

$$\mathrm{d}{f}_{\mid V_{\alpha }}=T^{*}\alpha \circ \mathrm{d}(f\circ {\alpha }^{-1})\circ \alpha :V_{\alpha }\to T^{*}{V}_{\alpha }$$

is a smooth map from $V_{\alpha }$ to $T^{*}{V}_{\alpha }$ . This holds for every chart $\alpha \in \mathfrak{A}(S)$ . Hence, $\mathrm{d}f:S\to T^{*}S$ is smooth. ■

Proposition 5.3. The space $\mathrm{Sec}{T}^{*}S$ of sections of $T^{*}S$ is closed under the operations of addition of sections and multiplication of a section by a smooth function.

Proof. In Proposition 4.6, we showed that, for every $\alpha \in \mathfrak{A}(S)$ , the space $\mathrm{Sec}{T}^{*}{W}_{\alpha }$ is closed under the operations of addition of sections and multiplication of a section by a smooth function. It follows from Definition 4.9 and Definition 4.10 that this property of sections extends to tangent bundles of locally Euclidean differential spaces. ■

Let $\vartheta :S\to T^{*}S$ be a section of $T^{*}S$ . For every $x\in S$ , there exists an open neighbourhood V of $x$ in S, $n\in \mathrm{\mathbb{N}}\mathrm{and}$ functions $e_1,\dots ,e_n,f_1,\dots ,f_n\in C^{\infty }(S)$ such that the restriction ${\vartheta }_{\mid V}:V\to T_V^{*}S=T^{*}S$ can be written in the following form:

$${\vartheta }_{\mid V}=\sum _{i=1}^n{p}_{i\mid V}\mathrm{d}{q}_{i\mid V}.$$

(65)

5.2. Canonical form in Koszul space $\mathrm{Der}{C}^{\infty }{(T^{*}S)}^{*}$

The canonical 1-form ${\theta }_M$ of the cotangent bundle $T^{*}M$ of a manifold $M$ was introduced by Yano and Patterson [18]. Here, we follow the approach of Abraham and Marsden [19]. ${\theta }_M$ is a linear map

$${\theta }_M:\mathrm{Der}{C}^{\infty }(T^{*}M)\to C^{\infty }(T^{*}M):Z\mapsto ⟨{\theta }_M\mid Z⟩$$

(66)

such that, for every vector field Z on $T^{*}M$ , and every $\mathrm{d}{f}_{\mid x}\in T_x^{*}M$ ,

$$⟨{\theta }_M\mid Z⟩(\mathrm{d}{f}_{\mid x})=⟨\mathrm{d}{f}_{\mid x}\mid T{\pi }_M(Z(\mathrm{d}{f}_{\mid x}))⟩.$$

(67)

But, $\mathrm{d}f:M\to T^{*}M:x\mapsto \mathrm{d}{f}_{\mid x}$ is a smooth section of $T^{*}M$ , so that, for every $f\in C^{\infty }(M)$ and every $Z\in \mathrm{Der}(T^{*}M)$ ,

$$⟨{\theta }_M\mid Z⟩\circ \mathrm{d}f=⟨\mathrm{d}f\mid T{\pi }_M\circ Z\circ \mathrm{d}f⟩\in C^{\infty }(M)$$

Hence, for every section $\vartheta :M\to T^{*}M$ ,

$$⟨{\theta }_M\mid Z⟩\circ \vartheta =⟨\vartheta \mid T{\pi }_M\circ Z\circ \vartheta ⟩\in C^{\infty }(M)$$

(68)

The generalization of the notion of the canonical 1-form ${\theta }_M$ of the cotangent bundle $T^{*}M$ of a manifold M, given above, to the notion of canonical 1-form ${\theta }_S$ of the cotangent bundle $T^{*}S$ of a locally Euclidean differential space S is almost straightforward. On a manifold M, all derivations of $C^{\infty }(M)$ generate local one-parameter groups of local diffeomorphisms of M, i.e., $\mathrm{Der}{C}^{\infty }(T^{*}M)=\mathfrak{X}(M)$ , but it is not so for general differential spaces. Hence, we must choose which type of derivations we choose in our generalization. Here, we choose derivations in $\mathrm{Der}{C}^{\infty }(T^{*}S)$ , which leads us to Koszul forms in $\wedge \mathrm{Der}{C}^{\infty }{(T^{*}S)}^{*}$ discussed in the work by Śniatycki [13] and reviewed in Appendix 1. The choice leading to Koszul forms in $\wedge {\mathfrak{X}}^{*}(S)$ was discussed in the work by Bates et al. [11].

Since all the maps used above extend to differential spaces, we adopt the following definition of the canonical Koszul 1-form ${\theta }_S$ in $\mathrm{Der}{C}^{\infty }{(T^{*}S)}^{*}$ of the cotangent bundle $T^{*}S$ . For the sake of simplicity, we omit in the following the terms specifying the space of forms used and will write the canonical 1-form ${\theta }_S$ .

Definition 5.4. The canonical 1-form ${\theta }_S$ of the cotangent space $T^{*}S$ of a locally Euclidean differential space S is a linear map

$${\theta }_S:\mathrm{Der}{C}^{\infty }(T^{*}S)\to C^{\infty }(T^{*}S):Z\mapsto ⟨{\theta }_S\mid Z⟩$$

(69)

such that

$$⟨{\theta }_S\mid Z⟩(\mathrm{d}{f}_{\mid x})=⟨\mathrm{d}{f}_{\mid x}\mid T{\pi }_S\circ Z(\mathrm{d}{f}_{\mid x})⟩$$

(70)

for every $\mathrm{d}{f}_{\mid x}\in T^{*}S$ . Hence, for every section $\vartheta :S\to T^{*}S$ ,

$$⟨{\theta }_S\mid Z⟩\circ \vartheta =⟨\vartheta \mid T{\pi }_S\circ Z\circ \vartheta ⟩\in C^{\infty }(S).$$

(71)

We denote by $\mathrm{d}{\theta }_S$ the exterior differential of ${\theta }_S$ and by $Z˩(\mathrm{d}{\theta }_S)$ the left interior product of the form $\mathrm{d}{\theta }_S$ by a derivation Z. ⁸

Definition 5.5. The kernel of $\mathrm{d}{\theta }_S$ is

$$\ker \mathrm{d}{\theta }_S=\{Z\in \mathrm{Der}{C}^{\infty }(T^{*}S)\mid Z˩(\mathrm{d}{\theta }_S)=0\},$$

(72)

If $\ker \mathrm{d}{\theta }_S=0$ , then $\mathrm{d}{\theta }_S$ is a symplectic form on $T^{*}S$ and is called the canonical symplectic form of $T^{*}S$ .

Remark 5.6. If $\ker \mathrm{d}{\theta }_S=0$ , for every $H\in C^{\infty }(T^{*}S)$ , then there exists a unique $Z_H\in \mathrm{Der}{C}^{\infty }(T^{*}S)$ such that

$$Z_H˩\mathrm{d}{\theta }_S=-\mathrm{d}H.$$

(73)

The derivation $Z_H$ satisfying equation (75) is referred to as the Hamiltonian derivation of H. ⁹ The space

$$\mathrm{Der}{C}^{\infty }{(T^{*}S)}^{\mathrm{Ham}}=\{Z_H\in \mathrm{Der}{C}^{\infty }(T^{*}S)\mid H\in C^{\infty }(T^{*}S)\}$$

of Hamiltonian derivations of $C^{\infty }(T^{*}S)$ is a Lie subalgebra of $\mathrm{Der}{C}^{\infty }(T^{*}S)$ .

The only difference between differential spaces and manifolds is that the Hamiltonian derivations need not generate local one-parameter groups of local diffeomorphisms of $T^{*}S$ . On the other hand, vector fields on $T^{*}S$ , i.e., derivations in $\mathfrak{X}(T^{*}S)$ , generate generate local one-parameter groups of local diffeomorphisms of $T^{*}S$ .

Definition 5.7. The space of Hamiltonian vector fields on $T^{*}S$ is

$$\mathfrak{X}{(T^{*}S)}^{\mathrm{Ham}}=\mathrm{Der}{C}^{\infty }{(T^{*}S)}^{\mathrm{Ham}}\cap \mathfrak{X}(T^{*}S)$$

(74)

of Hamiltonian derivations that generate local one-parameter groups of local diffeomorphisms of $T^{*}S$ .

The space $\mathfrak{X}{(T^{*}S)}^{\mathrm{Ham}}$ is a Lie subalgebra of $\mathfrak{X}(T^{*}S)$ , but it is not invariant under multiplication by functions in $C^{\infty }(T^{*}S)$ .

Definition 5.8. An (infinitesimal) symmetry of the canonical 1-form ${\theta }_S$ of $T^{*}S$ is a vector field $Z\in \mathfrak{X}(T^{*}S)$ , such that the Lie derivative ${\pounds }_Z{\theta }_S$ of ${\theta }_S$ with respect to Z vanishes.

Since

$${\pounds }_Z{\theta }_S=Z˩(\mathrm{d}{\theta }_S)+\mathrm{d}⟨{\theta }_S\mid Z⟩,$$

(75)

the condition that $Z\in \mathrm{Der}{C}^{\infty }(T^{*}S)$ is a symmetry of ${\theta }_S$ is

$$Z˩(\mathrm{d}{\theta }_S)=-\mathrm{d}⟨{\theta }_S\mid Z⟩.$$

(76)

Since $⟨{\theta }_S\mid Z⟩\in C^{\infty }(T^{*}S)$ , it follows that symmetries of ${\theta }_S$ are Hamiltonian derivations on $T^{*}S$ . Moreover, $Z_H\in \mathrm{Der}{C}^{\infty }{(T^{*}S)}^{\mathrm{Ham}}$ is a symmetry of ${\theta }_S$ if $H=⟨{\theta }_S\mid Z_H⟩$ .

Remark 5.9.

• (i) We have no example of a differential space S with $\ker \mathrm{d}{\theta }_S\ne 0$ , and no proof that $\ker \mathrm{d}{\theta }_S=0$ for every locally Euclidean differential space S.

  In the following, we continue our investigation of the structure of the cotangent bundle $T^{*}S$ of a locally Euclidean differential space S under assumption that $\ker \mathrm{d}{\theta }_S=0$ .

Consider $Z\in \mathrm{Der}{C}^{\infty }(T^{*}S)$ , which is a symmetry of ${\theta }_S$ . Equation (71) implies that if Z is tangent to the fibres of the cotangent bundle projection ${\pi }_S:T^{*}S\to S$ , then $\mathrm{T\pi }\circ Z=0$ , so that $\mathrm{d}⟨{\theta }_S\mid Z⟩=0$ . Since we assume that $\ker \mathrm{d}{\theta }_S=0$ , it follows that $⟨{\theta }_S\mid Z⟩=0$ implies that $Z=0$ . Hence, a symmetry $Z\in \mathrm{Der}{C}^{\infty }(T^{*}S)$ of ${\theta }_S$ is uniquely determined by its projection $\mathrm{T\pi }\circ Z:T^{*}S\to \mathrm{TS}$ .

6. Partition of $T^{*}S$

The partition $\mathfrak{M}(S)$ of a locally Euclidean differential space S by orbits of the Lie algebra $\mathfrak{X}(S)$ gives rise to a partition of its cotangent bundle

$$T^{*}S={\coprod }_{M\in \mathfrak{M}(S)}{T}_M^{*}S,$$

(77)

where $T_M^{*}S={\pi }_S^{-1}(M)$ and ${\pi }_S:T^{*}S\to S$ is the cotangent bundle projection.

For $M\in \mathfrak{M}(S)$ , let ${\iota }_M:M\to S$ denote the inclusion map. In the following presentation, we usually identify M with ${\iota }_M(M)\subseteq S$ and write ${\iota }_M(M)=M$ . We denote by ${\widehat{\pi }}_M:T_M^{*}S\to M$ the restriction of the cotangent bundle projection ${\pi }_S:T^{*}S\to S$ to codomain M. For every $\mathrm{d}{f}_{\mid x}\in T_M^{*}S$ ,

$${\widehat{\pi }}_M(\mathrm{d}{f}_{\mid x})={\pi }_S(\mathrm{d}{f}_{\mid x})=x\in M.$$

(78)

It is clear that fibres of ${\widehat{\pi }}_M:T_M^{*}S\to M$ are vector spaces.

Proposition 6.1. For every $e\in C^{\infty }(S)$ and $t\in \mathrm{ℝ}$ , the map

$${\mathrm{\Phi }}_t^e:T^{*}S\to T^{*}S:\mathrm{d}{f}_{\mid x}\mapsto \exp (t\mathrm{d}e)(\mathrm{d}{f}_{\mid x})=\mathrm{d}{f}_{\mid x}+t\mathrm{d}{e}_{\mid x}$$

(79)

is a one-parameter group of diffeomorphisms of $T^{*}S$ , which act linearly in fibres of ${\pi }_S:T^{*}S\to S$ .

Proof. Clearly, ${\mathrm{\Phi }}_t^e$ is a one-parameter group of transformations of $T^{*}S$ , which acts linearly in the fibres of the cotangent bundle projection ${\pi }_S:T^{*}S\to S$ . It follows from Definition 4.10 that in order to establish smoothness of ${\mathrm{\Phi }}_t^e:T^{*}S\to T^{*}S$ , it suffices to show that Proposition 6.1 holds whenever $S=W\subseteq {\mathrm{ℝ}}^n$ is a differential subspace of ${\mathrm{ℝ}}^n$ .

In Theorem 4.2, we showed that the musical maps ${\mathit{g}}_{\mathrm{TW}}^{♭}:\mathrm{TW}\to T^{*}W$ and ${\mathit{g}}_{T^{*}W}^{♯}:T^{*}W\to \mathrm{TW}$ , defined by the Euclidean metric of ${\mathrm{ℝ}}^n$ , are diffeomorphisms. In Theorem 4.4, we showed that, for every $e\in C^{\infty }(W)$ , the differential $\mathrm{d}e:W\to T^{*}W:\mathit{x}\mapsto \mathrm{d}{e}_{\mid \mathit{x}}$ is smooth. Hence, the gradient of e, given by

$$\mathrm{grad}e={\mathit{g}}_{T^{*}W}^{♯}\circ \mathrm{d}e:W\to \mathrm{TW}:\mathit{x}\mapsto {\mathit{g}}_{T^{*}W}^{♯}(\mathrm{d}{e}_{\mid \mathit{x}})$$

(80)

is a smooth section of the tangent bundle projection ${\tau }_W:\mathrm{TW}\to W$ . The differential structure $C^{\infty }(W)$ is generated by the restriction to W of smooth functions on ${\mathrm{ℝ}}^n$ . Suppose that there exists $E\in C^{\infty }({\mathrm{ℝ}}^n)$ such that $e=E_{\mid W}$ . Then,

$$\mathrm{grad}E={\mathit{g}}^{♯}\circ \mathrm{d}E:{\mathrm{ℝ}}^n\to T{\mathrm{ℝ}}^n:\mathit{x}\mapsto {\mathit{g}}^{♯}(\mathrm{d}{E}_{\mid \mathit{x}})$$

(81)

is a vector field on ${\mathrm{ℝ}}^n$ , which restricts to $\mathrm{grad}e$ in W. That is,

$$\mathrm{grad}{E}_{\mid W}=grade.$$

(82)

The function $E\in C^{\infty }({\mathrm{ℝ}}^n)$ gives rise to a one-parameter group

$${\mathrm{\Psi }}_t^E:T{\mathrm{ℝ}}^n\to T{\mathrm{ℝ}}^n:{\mathrm{v}}_{\mathit{x}}\mapsto \exp (t\mathrm{grad}E)({\mathrm{v}}_{\mathit{x}})={\mathrm{v}}_{\mathit{x}}+t\mathrm{grad}{E}_{\mid \mathit{x}}$$

(83)

of diffeomorphisms of $T{\mathrm{ℝ}}^n$ , which preserves each fibre of the tangent bundle projection ${\tau }_{{\mathrm{ℝ}}^n}:T{\mathrm{ℝ}}^n\to {\mathrm{ℝ}}^n$ . It restricts to a one-parameter group

$${\mathrm{\Psi }}_t^e:\mathrm{TW}\to \mathrm{TW}:{\mathrm{v}}_{\mathit{x}}\mapsto \exp (t\mathrm{grad}e)({\mathrm{v}}_{\mathit{x}})={\mathrm{v}}_{\mathit{x}}+t\mathrm{grad}{e}_{\mid \mathit{x}}$$

(84)

of diffeomorphisms of TW, which preserves each fibre of the tangent bundle projection ${\tau }_W:\mathrm{TW}\to W$ . Moreover, the musical maps ${\mathit{g}}_{\mathrm{TW}}^{♭}$ and ${\mathit{g}}_{T^{*}W}^{♯}$ intertwine the action of ${\mathrm{\Psi }}_t^e$ on TW and the action of ${\mathrm{\Phi }}_t^e$ on $T^{*}W:$

$$\begin{array}{ccccc}\multicolumn{1}{c}{}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{\exp (t\mathrm{grad}e)}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{\mathrm{ℝ}\times \mathrm{TW}}& \multicolumn{1}{c}{\to }& \multicolumn{1}{c}{\mathrm{TW}}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{{\mathrm{id}}_{\mathrm{ℝ}}\times {\mathit{g}}_{\mathrm{TW}}^{♭}}& \multicolumn{1}{c}{\downarrow }& \multicolumn{1}{c}{}& \multicolumn{1}{c}{\downarrow }& \multicolumn{1}{c}{{\mathit{g}}_{\mathrm{TW}}^{♭}}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{\mathrm{ℝ}\times T^{*}W}& \multicolumn{1}{c}{\to }& \multicolumn{1}{c}{T^{*}W}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{\exp (t\mathrm{d}e)}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}\end{array}.$$

(85)

Even if $e\in C^{\infty }(W)$ is not the restriction to W of a single function on ${\mathrm{ℝ}}^n$ , for each point $\mathit{x}\in W$ , there exists an open subset $U_{\mathit{x}}$ of ${\mathrm{ℝ}}^n$ and a function $E_{\mathit{x}}\in C^{\infty }({\mathrm{ℝ}}^n)$ such that $e_{\mid U_{\mathit{x}}\cap W}=E_{\mathit{x}\mid U_{\mathit{x}}\cap W}$ . Hence,

$$\mathrm{grad}{e}_{\mid U_{\mathit{x}}\cap W}=grad{E}_{\mathit{x}\mid U_{\mathit{x}}\cap W},$$

which ensures that grad e is a vector field on TW. Since ${\mathit{g}}_{\mathrm{TW}}^{♭}:\mathrm{TW}\to T^{*}W$ is a diffeomorphism, it follows that it intertwines the action of ${\mathrm{\Psi }}_t^e$ on TW and the action of ${\mathrm{\Phi }}_t^e$ on $T^{*}W$ . In any case, ${\mathrm{\Phi }}_t^e$ is a one-parameter group of diffeomorphisms of $T^{*}S$ , which act linearly in fibres of ${\pi }_W:T^{*}W\to W$ . In view of the remark at the beginning of the proof, the proof is completed. ■

For every $e\in C^{\infty }(S)$ , we denote by $Z^e$ the vector field on $T^{*}S$ generating the one-parameter group

$${\mathrm{\Phi }}_t^e:T^{*}S\to T^{*}S:\mathrm{d}{f}_x\mapsto \mathrm{d}{f}_{\mid x}+t\mathrm{d}{e}_{\mid x}$$

of diffeomorphisms of $T^{*}S$ , which act linearly in fibres of ${\pi }_S:T^{*}S$ . The vector fields $Z^e$ are tangent to the fibres of the cotangent bundle projection ${\pi }_S:T^{*}S\to S$ . Hence, $T{\pi }_S\circ Z^e=0$ for every $e\in C^{\infty }(S)$ .

Let $X\in \mathfrak{X}(S)$ be vector field on S. It generates a local one-parameter group ${\mathrm{e}}^{\mathrm{tX}}:x\mapsto {\mathrm{e}}^{\mathrm{tX}(x)}$ of local diffeomorphisms of S. For every $x_0\in S$ , there exists an open neighbourhood $V_0$ of $x_0$ in S and $ϵ>0$ such that ${\mathrm{e}}_{\mid V_0}^{\mathrm{tX}}:V_0\to {\mathrm{e}}^{\mathrm{tX}}(V_0)$ is a diffeomorphism for $\left|t\right|$ $<ϵ$ . It follows from Remark 4.8 and Definition 4.10 that the cotangent lift

$$T^{*}{\mathrm{e}}_{\mid V_0}^{\mathrm{tX}}:T^{*}{V}_0\to T^{*}{\mathrm{e}}^{\mathrm{tX}}(V_0):\mathrm{d}{f}_{\mid {\mathrm{e}}^{\mathrm{tX}(x)}}\mapsto \mathrm{d}(f\circ {\mathrm{e}}^{\mathrm{tX}(x)})$$

is a diffeomorphism for every $\left|t\right|$ $<ϵ$ . Hence,

$$\begin{array}{ccc}\multicolumn{1}{c}{T^{*}{\mathrm{e}}^{\mathrm{tX}}}& \multicolumn{1}{c}{:}& \multicolumn{1}{c}{T^{*}S\to T^{*}S:\mathrm{d}{f}_{\mid {\mathrm{e}}^{\mathrm{tX}(x)}}\mapsto T^{*}{\mathrm{e}}^{\mathrm{tX}}(\mathrm{d}{f}_{\mid {\mathrm{e}}^{\mathrm{tX}(x)}})}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{=}& \multicolumn{1}{c}{\mathrm{d}{f}_{\mid {\mathrm{e}}^{\mathrm{tX}(x)}}\circ T{\mathrm{e}}^{\mathrm{tX}(x)}=\mathrm{d}(f\circ {\mathrm{e}}^{\mathrm{tX}(x)})=\mathrm{d}{(f\circ {\mathrm{e}}^{\mathrm{tX}})}_{\mid x}}& \multicolumn{1}{c}{}\end{array}$$

(86)

is a local one-parameter group of local diffeomorphisms of $T^{*}S$ , such that the following diagram commutes:

$$\begin{array}{ccccc}\multicolumn{1}{c}{}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{T^{*}{\mathrm{e}}^{\mathrm{tX}}}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{T^{*}S}& \multicolumn{1}{c}{\to }& \multicolumn{1}{c}{T^{*}S}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{{\pi }_S}& \multicolumn{1}{c}{\downarrow }& \multicolumn{1}{c}{}& \multicolumn{1}{c}{\downarrow }& \multicolumn{1}{c}{{\pi }_S}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{S}& \multicolumn{1}{c}{\leftarrow }& \multicolumn{1}{c}{S}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{{\mathrm{e}}^{\mathrm{tX}}}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}\end{array}.$$

(87)

We denote by $Z^X$ the vector field on $T^{*}S$ generating the $T^{*}{\mathrm{e}}^{\mathrm{tX}}:T^{*}S\to T^{*}S.$ That is, for every $h\in C^{\infty }(T^{*}S)$ ,

$$Z^{X}h=\frac{d}{\mathrm{dt}}{(h\circ T^{*}{\mathrm{e}}^{\mathrm{tX}})}_{\mid t=0}.$$

(88)

Commutativity of diagram (87) implies that, for every $\mathrm{d}{f}_{\mid x}\in T^{*}S$ ,

$$T{\pi }_S(Z^X(\mathrm{d}{f}_{\mid x}))=X(x).$$

(89)

Definition 6.2. Let

$$\mathcal{Z}=\{Z^e,Z^X\in \mathfrak{X}(T^{*}S)\mid X\in \mathfrak{X}(S),e\in C^{\infty }(S)\}$$

(90)

be the family of vector fields on $T^{*}S$ consisting of the generators $Z^e$ of one-parameter groups ${\mathrm{\Phi }}_t^e:T^{*}S\to T^{*}S$ , for $e\in C^{\infty }(S)$ , and the generators $Z^X$ of local one parameter local groups $T^{*}{\mathrm{e}}^{\mathrm{tX}}:T^{*}S\to T^{*}S$ , for $X\in \mathfrak{X}(S)$ .

Theorem 6.3.

• (i) The members of the partition $T^{*}S={\coprod }_{M\in \mathfrak{M}(S)}{T}_M^{*}S$ under consideration are orbits of the family

  $$\mathcal{Z}=\{Z^e,Z^X\in \mathfrak{X}(T^{*}S)\mid X\in \mathfrak{X}(S),e\in C^{\infty }(S)\}$$

  of vector fields on $T^{*}S$ .

• (ii)  $T_M^{*}S$ is a submanifold of $T^{*}S$ for every $M\in \mathfrak{M}(S)$ .

Proof.

• (i) Since every ${\mathrm{\Phi }}^e$ preserves the fibres of ${\pi }_S:T^{*}S\to S$ , it preserves $T_M^{*}S$ for every $M\in \mathfrak{M}(S)$ . Similarly, for every $X\in \mathfrak{X}(S)$ , the local diffeomorphism ${\mathrm{e}}^{\mathrm{tX}}$ preserves every $M\in \mathfrak{M}(S)$ . Hence, for every $M\in \mathfrak{M}(S)$ , the differential subspace $T_M^{*}S$ of $T^{*}S$ is preserved by all ${\mathrm{\Phi }}^e$ , for $e\in C^{\infty }(S)$ , and by all $T^{*}{\mathrm{e}}^{\mathrm{tX}}$ , for $X\in \mathfrak{X}(S)$ . Therefore, the orbits of $\mathcal{Z}$ through points in $T_M^{*}S$ are contained in $T_M^{*}S$ .

  On the other hand, the family of transformations $\{{\mathrm{\Phi }}^e\mid e\in C^{\infty }(S)\}$ acts transitively on every fibre $T_x^{*}S\subseteq T^{*}S$ , and the family of transformations $\{T^{*}{\mathrm{e}}^{\mathrm{tX}}\mid X\in \mathfrak{X}(S)\}$ acts transitively on the space of fibres $T_x^{*}S\subseteq T_M^{*}S$ because $\{{\mathrm{e}}^{\mathrm{tX}}$ $\mid X$ $\in \mathfrak{X}(S)\}$ acts transitively on every $M\in \mathfrak{M}(S)$ . Therefore, every $T_M^{*}S$ , for $M\in \mathfrak{M}(S)$ , is an orbit of $\{{\mathrm{\Phi }}^k,T^{*}{\mathrm{e}}^{\mathrm{tX}}\mid k\in C^{\infty }(S)$ and $X\in \mathfrak{X}(S)\}$ . Since $\mathcal{Z}$ generates $\{{\mathrm{\Phi }}^k,T^{*}{\mathrm{e}}^{\mathrm{tX}}\mid k\in C^{\infty }(S)$ and $X\in \mathfrak{X}(S)\}$ , it follows that every $T_M^{*}S$ is an orbit of $\mathcal{Z}$ .

• (ii) By Theorem 3.4.5 of Śniatycki [13], every orbit of a family of vector fields on a locally Euclidean differential space is a submanifold of the ambient space ¹⁰ . Since $\mathcal{Z}$ is a family of vector fields on $T^{*}S$ , and its orbits are the parts $T_M^{*}S$ of the partition of $T^{*}S$ under consideration, it follows that $T_M^{*}S$ is a submanifold of $T^{*}S$ for every $M\in (S)$ . ■

Definition 6.4. For every $M\in \mathfrak{M}(S)$ , we denote by ${\theta }_{T_M^{*}S}$ and ${ɷ}_{T_M^{*}S}$ the pull-backs to $T_M^{*}S$ of ${\theta }_S$ and $\mathrm{d}{\theta }_S$ , respectively, so that

$${ɷ}_{T_M^{*}S}=\mathrm{d}{\theta }_{T_M^{*}S}.$$

(91)

Also,

$$\ker {ɷ}_{T_M^{*}S}=\ker \mathrm{d}{\theta }_{T_M^{*}S}=\{Z\in \mathrm{Der}{C}^{\infty }(T_M^{*}S)\mid Z˩(\mathrm{d}{\theta }_{T_M^{*}S})=0\}.$$

(92)

Let $X\in \mathfrak{X}(S)$ be a vector field on S. It gives rise to a smooth function on $T^{*}S$ , also denoted X, ¹¹

$$X:T^{*}S\to \mathrm{ℝ}:\mathrm{d}{f}_{\mid x}\mapsto ⟨\mathrm{d}{f}_{\mid x}\mid X⟩\in C^{\infty }(T^{*}S).$$

(93)

The assumption that $\ker \mathrm{d}{\theta }_S=0$ implies that there exists a unique Hamiltonian derivation $Z_X$ corresponding to the Hamiltonian $H=X$ . That is, for every $\mathrm{d}{f}_{\mid x}\in T^{*}S$ ,

$$H(\mathrm{d}{f}_{\mid x})=X(\mathrm{d}{f}_{\mid x})=⟨\mathrm{d}{f}_{\mid x}\mid X⟩.$$

Moreover, $Z_X˩\mathrm{d}{\theta }_S=-\mathrm{d}⟨\mathrm{d}{f}_{\mid x}\mid X⟩$ . Hence, for every $e\in C^{\infty }(S)$ ,

$$\begin{array}{cccc}\multicolumn{1}{c}{}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{⟨\mathrm{d}{\theta }_S\mid (Z_X,Z^e)⟩(\mathrm{d}{f}_{\mid x})=-Z^e˩\mathrm{d}⟨\mathrm{d}{f}_{\mid x}\mid X⟩}& \multicolumn{1}{c}{}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{=}& \multicolumn{1}{c}{\frac{d}{\mathrm{dt}}{⟨\mathrm{d}{f}_{\mid x}+t\mathrm{d}{e}_{\mid x}\mid X⟩}_{\mid t=0}=⟨\mathrm{d}{e}_{\mid x}\mid X(x)⟩.}& \multicolumn{1}{c}{}\end{array}$$

It follows that, for every $\mathrm{d}{f}_{\mid x}\in T^{*}S$ ,

$$T{\pi }_S(Z_X(\mathrm{d}{f}_{\mid x}))=X(x).$$

(94)

Moreover, for every $X\in \mathfrak{X}(S)$ and every $e\in C^{\infty }(S)$ ,

$$⟨\mathrm{d}{\theta }_S\mid (Z_X,Z^e)⟩=⟨\mathrm{d}e\mid X⟩.$$

(95)

For $e,f\in C^{\infty }(S)$ , it follows from equation (100)

$$⟨\mathrm{d}{\theta }_S\mid (Z^e,Z^f)⟩=Z^e⟨{\theta }_S\mid Z^f⟩-Z^f⟨{\theta }_S\mid Z^e⟩-⟨{\theta }_S\mid [Z^e,Z^f]⟩.$$

It is easy to see that $[Z^e,Z^f]=0$ . Moreover, equation (71) yields $⟨{\theta }_S\mid Z^e⟩=⟨{\theta }_S\mid Z^f⟩=0$ . Hence,

$$⟨\mathrm{d}{\theta }_S\mid (Z^e,Z^f)⟩=0$$

(96)

for every $e,f\in C^{\infty }(S)$ .

Theorem 6.5. Let S be a locally Euclidean differential space such that $\ker$ $d{\theta }_S=0$ .

• (i) For every $M\in \mathfrak{M}(S)$ ,

  $$\ker \mathrm{d}{\theta }_{T_M^{*}S}=\{Z_{\mid \mathrm{d}{f}_{\mid x}}^e\in T(T_M^{*}S)\mid \mathrm{d}{f}_{\mid x}\in T_M^{*}S,e_{\mid M}=\mathrm{constant}\}.$$

  (97)

• (ii) Moreover,

  $$\begin{array}{cc}\multicolumn{1}{c}{\{\mathrm{d}{e}_{\mid x}\in }& \multicolumn{1}{c}{T_M^{*}S\mid x\in M,e\in C^{\infty }(S)\mathrm{and}{e}_{\mid M}=\mathrm{constant}\}=}\\ \multicolumn{1}{c}{}& \multicolumn{1}{c}{=\mathrm{ATM}=\{\mathrm{d}{f}_{\mid x}\in T_M^{*}S\mid ⟨\mathrm{d}{f}_{\mid x}\mid \mathrm{v}⟩=0\forall \mathrm{v}\in T_{x}M\}.}& \multicolumn{1}{c}{}\end{array}$$

  (98)

• (iii) Consequently, $(T_M^{*}S$ , $d{\theta }_{T_M^{*}S})$ is a presymplectic manifold with reduced symplectic space

  $$(T_M^{*}S∕\ker \mathrm{d}{\theta }_{T_M^{*}S},\mathrm{d}{\theta }_{T_M^{*}S\mid }∕\ker \mathrm{d}{\theta }_{T_M^{*}S})=(T^{*}M,{ɷ}_M),$$

  (99)

  where ${ɷ}_M=$ $d{\theta }_M$ is the canonical symplectic form of the cotangent bundle $T^{*}M$ of the manifold $M\in \mathfrak{M}(S)$ .

Proof.

• (i) It follows from Theorem 6.3 that $T_M^{*}S$ is a submanifold of $T^{*}S$ . In particular, $T(T_M^{*}S)$ is spanned by the restrictions to $T_M^{*}S$ of the family

  $$\mathcal{Z}=\{Z^e,Z^X\in \mathfrak{X}(T^{*}S)\mid X\in \mathfrak{X}(S),e\in C^{\infty }(S)\}$$

  of vector fields on $T^{*}S$ . In fact,

  $$T(T_M^{*}S)=\{Z^e(\mathrm{d}{f}_{\mid x}),Z^X(\mathrm{d}{f}_{\mid x})\mid \mathrm{d}{f}_{\mid x}\in T_M^{*}S,X\in \mathfrak{X}(M),e\in C^{\infty }(S)\}.$$

  For every $X\in \mathfrak{X}(S)$ , and every $\mathrm{d}{f}_{\mid x}\in T^{*}S$ , equations (89) and (94) yield

  $$T{\pi }_S(Z^X(\mathrm{d}{f}_{\mid x}))=T{\pi }_S(Z_X(\mathrm{d}{f}_{\mid x}))=X(x).$$

  Hence, in $\mathcal{Z}$ , we may replace $Z^X$ by $Z_X$ and obtain the family

  $${\mathcal{Z}}^{\prime }=\{Z^e,Z_X,\in \mathrm{Der}{C}^{\infty }(T^{*}S)\mid X\in \mathfrak{X}(S),e\in C^{\infty }(S)\}$$

  of derivations of $C^{\infty }(T^{*}S)$ , such that

  $$T(T_M^{*}S)=\{Z^e(\mathrm{d}{f}_{\mid x}),Z_X(\mathrm{d}{f}_{\mid x})\mid \mathrm{d}{f}_{\mid x}\in T_M^{*}S,X\in \mathfrak{X}(M),e\in C^{\infty }(S)\}.$$

  Equations (95) and (96) show that $X^e\in {\mathcal{Z}}^{\prime }$ restricted to $T_M^{*}S$ is in $\ker {ɷ}_{T_M^{*}S}$ if $e_{\mid M}=\mathrm{const},$ so that $\mathrm{d}{e}_{\mid M}=0$ . If $\mathrm{d}{e}_{\mid M}\ne 0$ , then there exists $X\in \mathfrak{X}(M)$ , such that ${⟨\mathrm{d}{\theta }_S\mid Z_X,Z^e⟩}_{\mid T_M^{*}S}=⟨\mathrm{d}{e}_{\mid M}\mid X⟩\ne 0$ . Hence, $\ker$ $d{\theta }_{T_M^{*}S}=\{Z_{\mid T_M^{*}S}^e\mid$ $d{e}_{\mid M}=0\}$ , which completes the proof of (i).

• (ii) For every $d{e}_{\mid x}\in T_M^{*}S$ and every $\mathrm{v}\in T_{x}M$ , $⟨\mathrm{d}e\mid \mathrm{v}⟩$ $=0$ if and only if $e_{\mid M}=\mathrm{constant}$ .

• (iii) In Definition 4.1, we identified $T^{*}W$ with $T_W^{*}{\mathrm{ℝ}}^n∕\mathrm{ATW}$ in order to determine the differential structure of $T^{*}W$ . Similarly, here the cotangent map

  $$T^{*}{\iota }_M:T_M^{*}S\to T^{*}M:\mathrm{d}{f}_{\mid x}\mapsto T^{*}{\iota }_M(\mathrm{d}{f}_{\mid x})=\mathrm{d}{({\iota }_M^{*}f)}_{\mid \mathit{x}}$$

  associates with every $\mathrm{d}{f}_{\mid x}\in T_M^{*}S$ the covector $T^{*}{\iota }_M(\mathrm{d}{f}_x)$ such that, for every $\mathrm{v}\in T_{x}M\subseteq T_{x}S$ ,

  $$⟨T^{*}{\iota }_M(\mathrm{d}{f}_{\mid x})\mid \mathrm{v}⟩=⟨\mathrm{d}{f}_{\mid \mathit{x}}\mid T{\iota }_W(\mathrm{v})⟩,$$

As in section 4.1, this argument leads to

$$T^{*}M=T^{*}{\iota }_M(T_M^{*}S)=T_M^{*}S∕\mathrm{ATM},$$

where

$$\mathrm{ATM}=\{d{f}_{\mid x}\mid x\in M\mathrm{and}⟨\mathrm{d}{f}_{\mid x}\mid \mathrm{v}⟩=0\mathrm{for}\mathrm{all}\mathrm{v}\in T_{x}M\}.$$

Taking into account part (ii) of the theorem, we get

$$T_M^{*}S∕\ker \mathrm{d}{\theta }_{T_M^{*}S}=T_M^{*}S∕\mathrm{ATM}=T^{*}M$$

and the identification

$${ɷ}_{T_M^{*}S\mid }∕\ker \mathrm{d}{\theta }_{T_M^{*}S}=\mathrm{d}{\theta }_{T_M^{*}S\mid }∕\ker \mathrm{d}{\theta }_{T_M^{*}S}={ɷ}_M=\mathrm{d}{\theta }_M.$$