Intrinsic Geometric Structure of Subcartesian Spaces

Abstract

Every subset S of a Cartesian space ${\mathbb{R}}^d$, endowed with differential structure $C^{\infty }\left(S\right)$ generated by restrictions to S of functions in $C^{\infty }\left({\mathbb{R}}^d\right),$ has a canonical partition $\mathfrak{M}\left(S\right)$ by manifolds, which are orbits of the family $\mathfrak{X}\left(S\right)$ of all derivations of $C^{\infty }\left(S\right)$ that generate local one-parameter groups of local diffeomorphisms of S. This partition satisfies the frontier condition, Whitney’s conditions A and B. If $\mathfrak{M}\left(S\right)$ is locally finite, then it satisfies all definitions of stratification of S. This result extends to Hausdorff locally Euclidean differential spaces. The partition $\mathfrak{M}\left(S\right)$ of a subcartesian space S by smooth manifolds provides a measure for the applicability of differential geometric methods to the study of the geometry of S. If all manifolds in $\mathfrak{M}\left(S\right)$ are single points, we cannot expect differential geometry to be an effective tool in the study of S. On the other extreme, if $\mathfrak{M}\left(S\right)$ contains only one manifold M, then the subcartesian space S is a manifold, $S=M,$ and it is a natural domain for differential geometric techniques.

1. Introduction

In the second half of the twentieth century, the idea of using differential geometry to study spaces with singularities was in the air. In 1955, Satake introduced the notion of a V manifold in terms of an atlas of charts with values in quotients of connected open subsets of ${\mathbb{R}}^n$ by a finite group of linear transformations [1].

In 1961, Cerf [2] introduced the notion of generalized manifold, now known as manifold with corners, defined in terms of an atlas of charts with values in open subsets of ${[0,\infty )}^k\times {\mathbb{R}}^{n-k}\subseteq {\mathbb{R}}^n$, where $k=0,1,\dots n$. Cerf had all elements of the definition of the general class of differential spaces, but he did not develop the corresponding general theory. He preferred to investigate the example provided by manifolds with corners.

In 1966, Smith [3] introduced his notion of differentiable structure on a topological space which consists of a family of continuous functions on the space, deemed to be smooth and which carry all the information about the geometry of the space. Smith used the term differentiable spaces, and he studied the de Rham theorem on differentiable spaces.

In 1967, Sikorski [4] generalized Smith’s approach and used it to discuss the notion of an abstract covariant derivative. Sikorski used the term differential structure for the collection of functions on a topological space deemed to be smooth, and the term differential space for a topological space endowed with a differential structure. In 1974, Sikorski [5] published a book on differential geometry, in which he started with the development of the theory of differential spaces and later specified the spaces under consideration to be smooth manifolds. Sikorski used his book as the text in his masters-level course in differential geometry at the University of Warsaw. Even though Sikorski’s book was written in Polish, it was appreciated by a sizeable group of international scientists. Also in 1967, Aronszajn [6] introduced, in the abstract to his presentation at a meeting of the American Mathematical Society, the notion of a subcartesian space as a Hausdorff topological space that is locally diffeomorphic to a subset of a Cartesian (Euclidean) space. The local diffeomorphisms used by Aronszajn formed an atlas, similar to that introduced by Cerf. A more comprehensive presentation of this theory and its applications were given by Aronszajn and Szeptycki in 1975 [7] and in 1980 [8], and by Marshall in 1975 [9].

There are other theories allowing for the study of differential geometry of singular spaces. For a more comprehensive review, see reference [10].

Here, we concentrate on the theories of Aronszajn and Sikorski. The strength of Aronszajn’s approach is his choice of assumptions, which are satisfied by most finite-dimensional examples. On the other hand, Sikorski made the weakest assumptions. It leads to the simplicity of the basic presentation of the theory and makes other theories into special cases of Sikorski’s theory of differential spaces. The relation between the theories of Aronszajn and Sikorski was discussed first by Walczak [11] in 1973. In 2021, we exhibited a natural transformation from the category of subcartesian spaces to the category of Hausdorff locally Euclidean differential spaces [12]. Since Hausdorff locally Euclidean differential spaces can be identified with corresponding subcartesian spaces, the terms Hausdorff locally Euclidean differential space and subcartesian space are treated here as synonyms and used interchangeably. Aronszajn’s term is shorter and well-known to experts, but it does not convey much information to the uninitiated. That is why we use the longer term in the abstract and explanations. In the proofs, we use the shorter term.

Section 2 contains a brief review of the elements of the theory of differential spaces, followed by references, that are essential for this paper. The presentation is based on the book by Sikorski [5].

A more comprehensive review of derivations of the differential structure of a differential space and their integration based on [13,14] is given in Section 3. We use the term vector fields on a subcartesian space $S$(Hausdorff locally Euclidean differential space) for derivations of ${\mathcal{C}}^{\infty }\left(S\right)$ that generate local one-parameter groups of local diffeomorphisms of S. Orbits of the family of all vector fields on a subcartesian space S form a partition $\mathfrak{M}\left(S\right)$ of S by smooth manifolds. This section contains two previously unpublished results with proofs.

Section 4 is devoted to the study of the partition $\mathfrak{M}\left(S\right)$ of a differential space S by orbits of the family of all vector fields on S, which is the main objective of this paper. In the case when the differential space under consideration is a connected manifold M, the Lie algebra of local one-parameter groups of local diffeomorphisms of M acts transitively of $M,$ which means that the corresponding partition of M is trivial, and it consists of a single orbit. We show that the partition $\mathfrak{M}\left(S\right)$ satisfies the frontier condition and Whitney’s conditions A and B.

In Section 5, we compare the results of Section 4 with various definitions of stratifications. If the partition $\mathfrak{M}\left(S\right)$ is locally finite, then it satisfies all definitions of a stratification of a closed subset of a smooth manifold.

In Section 6, we briefly relate derivations that are not vector fields to transient vector fields on manifolds with boundaries discussed by Percel [15]. These derivations generate transitions between different manifolds of the partition $\mathfrak{M}\left(S\right).$

In Section 7, we apply our approach to manifolds with corners. According to Cerf’s definition [2], a manifold with corners S is a locally closed subcartesian space. Following Joyce’s formulation of the theory of manifolds with corners [16], we show that the depth function stratification of S coincides with the partition $\mathfrak{M}\left(S\right)$, and it satisfies Whitney’s conditions A and B.

Following requests of the referees, we include a brief section with concluding remarks and an outline of the future research.

2. Differential Spaces

Definition 1.

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

1.

The family

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

is a sub-basis of the topology of S.

2.

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

3.

If $f:S\to \mathbb{R}$ is a function such that, for each $x\in S$, there is an open neighborhood V of x in S and a function $f_x\in C^{\infty }\left(S\right)$ such that its restriction $f_{x\mid V}$ to V is equal to the restriction $f_{\mid V}$ to V,

$${f_x}_{\mid V}=f_{\mid V},$$

then $f\in C^{\infty }\left(S\right)$.

A topological space S endowed with a differential structure $C^{\infty }\left(S\right)$ is called differential space.

Definition 2.

If S and R are differential spaces, endowed with differential structures $C^{\infty }\left(S\right)$ and $C^{\infty }\left(R\right)$, a map $\phi :S\to R$ is smooth if for each $f\in C^{\infty }\left(R\right)$ the pull back ${\phi }^{*}f=f \circ \phi$ is in $C^{\infty }\left(S\right)$. A smooth map $\phi :S\to R$ is a diffeomorphism if it is invertible and its inverse is smooth.

Note that if $(S,C^{\infty }\left(S\right))$ and $(R,C^{\infty }\left(R\right))$ are differential spaces, and a map $f:S\to R$ is smooth, then it is a homeomorphism of the underlying topological spaces. Differential spaces and smooth maps form a category.

A simple way of defining a differential structure on a set S is as follows. Choose a family of functions $\mathcal{F}$ on S. Endow S with the topology generated by a sub-basis

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

(1)

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

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

(2)

It is easy to see that the differential structure $C^{\infty }\left(S\right)$ generated by $\mathcal{F}$ satisfies all conditions of Definition 1.

3. Derivations and Vector Fields

In this section, we continue with the basic results required in this paper following references [13,14] and references within. We also include here a couple of new results with proofs.

Definition 3.

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

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

(3)

for every $f_1$, $f_2\in C^{\infty }\left(S\right)$.

Let $\mathrm{Der} C^{\infty }\left(S\right)$ denote the space of deriviations of $C^{\infty }\left(S\right)$. It is a Lie algebra with Lie bracket

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

(4)

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

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

(5)

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

Definition 4.

Let $\phi :R\to S$ be a smooth map of differential spaces with differential structures $C^{\infty }\left(R\right)$ and $C^{\infty }\left(S\right)$, respectively. Derivations X in $\mathrm{Der} C^{\infty }\left(S\right)$ and Y in $\mathrm{Der} C^{\infty }\left(R\right)$ are φ-related if

$${\phi }^{*}\left(Y\left(f\right)\right)=X\left({\phi }^{*}f\right)$$

for every $f\in C^{\infty }\left(R\right)$.

Suppose that the map $\phi :R\to S$ in Definition 4 is a diffeomorphism; that is, ${\phi }^{-1}:R\to S$ exists and is smooth. For every derivation $X\in \mathrm{Der} C^{\infty }\left(R\right)$, there exists a unique derivation ${\phi }_{*}X$$\in \mathrm{Der} C^{\infty }\left(S\right)$,

$${\phi }_{*}X:C^{\infty }\left(S\right)\to C^{\infty }\left(S\right):f\mapsto \left({\phi }_{*}X\right)f={\left({\phi }^{-1}\right)}^{*}\left(X\left({\phi }^{*}f\right)\right),$$

(6)

which is $\phi$-related to X. It is called the push-forward of X by $\phi$. Moreover,

$${\phi }_{*}:\mathrm{Der} C^{\infty }\left(R\right)\to \mathrm{Der} C^{\infty }\left(S\right):X\mapsto {\phi }_{*}X$$

is a Lie algebra diffeomorphism.

Definition 5.

Let S be a subcartesian space and X a derivation of ${\mathcal{C}}^{\infty }\left(S\right).$ 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 $\mathbb{R}$ containing 0, such that $c\left(0\right)=x_0$ and

$$\frac{df}{dt}\left(c\left(t\right)\right)=\left(Xf\right)\left(c\left(t\right)\right)\mathit{for}\mathit{every}f\in {\mathcal{C}}^{\infty }\left(S\right)\mathit{and}\mathit{every}t\in I,$$

(7)

whenever the interior of I is not empty.

Integral curves of a given derivation X of ${\mathcal{C}}^{\infty }\left(S\right)$ starting at $x_0$ can be ordered by the inclusion of their domains. In other words, if $c_1:I_1\to S$ and $c_2:I_2\to S$ are two integral curves of X, such that $c_1\left(0\right)=c_2\left(0\right)=x_0$, and $I_1\subseteq I_2$, then $c_1⪯c_2$. An integral curve $c:I\to S$ of X is maximal if $c⪯c_1$ implies that $c=c_1$.

Theorem 1.

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

Proof. 

See the proof of Theorem 3.2.1 in [14]. □

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

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

Proposition 1.

For every derivation X of the differential structure ${\mathcal{C}}^{\infty }\left(S\right)$ of a subcartesian space and a diffeomorphism $\phi :S\to R$,

$${\mathrm{e}}^{t{\phi }_{*}X}=\phi \circ {\mathrm{e}}^{tX}\circ {\phi }^{-1}.$$

Proof. 

For each $f\in {\mathcal{C}}^{\infty }\left(R\right)$ and $y=\phi \left(x\right)\in R,$

$$\begin{array}{ccc}& & \frac{d}{dt}f\left((\phi \circ {\mathrm{e}}^{tX}\circ {\phi }^{-1})\left(y\right)\right)=\frac{d}{dt}f(\phi \circ {\mathrm{e}}^{tX})\left(x\right)\hfill \\ & =& \left(T\phi \left(\frac{d}{dt}{\mathrm{e}}^{tX}\left(x\right)\right)\right)\left(f\right)=\left(\frac{d}{dt}{\mathrm{e}}^{tX}\left(x\right)\right)\left({\phi }^{*}f\right)\hfill \\ & =& X\left({\phi }^{*}f\right)\left({\mathrm{e}}^{tX}\left(x\right)\right)\mathrm{by}\mathrm{Equation}\left(7\right)\hfill \\ & =& {\phi }^{*}\left({\phi }_{*}X\left(f\right)\right)\left({\mathrm{e}}^{tX}\left(x\right)\right)\mathrm{by}\mathrm{Equation}\left(6\right)\hfill \\ & =& \left({\phi }_{*}X\left(f\right)\right)\left(\phi \left({\mathrm{e}}^{tX}\left(x\right)\right)\right)=\left({\phi }_{*}X\left(f\right)\right)(\phi \left({\mathrm{e}}^{tX}\left({\phi }^{-1}\left(y\right)\right)\right)\hfill \\ & =& \left({\phi }_{*}X\left(f\right)\right)(\phi \circ {\mathrm{e}}^{tX}\circ {\phi }^{-1})\left(y\right).\hfill \end{array}$$

Hence, $t\mapsto (\phi \circ {\mathrm{e}}^{tX}\circ {\phi }^{-1})\left(y\right)$ is an integral curve of ${\phi }_{*}X$ through y. □

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

Definition 6.

A vector field on a subcartesian space S is a derivation X of ${\mathcal{C}}^{\infty }\left(S\right)$ such that for every $x\in S$, there exists an open neighborhood U of x in S and $\epsilon >0$ such that for every $t\in (-\epsilon ,\epsilon ),$ the map ${\mathrm{e}}^{tX}\left(x\right)$ is defined on U, and its restriction to U is a diffeomorphism from U onto an open subset of S. In other words, X is a vector field on S if ${\mathrm{e}}^{tX}$ is a local one-parameter group of local diffeomorphisms of S. We denote by $\mathfrak{X}\left(S\right)$ the family of all vector fields on a subcartesian space S.

Theorem 2.

Let S be a subcartesian space. A derivation $X$ of ${\mathcal{C}}^{\infty }\left(S\right)$ is a vector field on S if the domain of every maximal integral curve of X is open in $\mathbb{R}$.

Proof. 

Suppose a derivation X of ${\mathcal{C}}^{\infty }\left(S\right)$ has a maximal integral curve of the type $c:\left\{0\right\}\to \left\{x\right\}:0\mapsto x$. This integral curve cannot generate a local one-parameter group of local diffeomorphisms of S. Hence, X is not a vector field. Therefore, in the remainder of the proof, we need not consider integral curves of this type. Consider the case when S is a differential subspace of ${\mathbb{R}}^n.$ Let X be a derivation of S such that domains of all its integral curves are open in $\mathbb{R}$. In other words, for each $x\in S$, the domain $I_x$ of the map $t\mapsto {\mathrm{e}}^{tX}\left(x\right)$ is an open interval in $\mathbb{R}$.

This implies that no maximal integral curve of X is defined only for $t=0$. We need to show that the map $x\mapsto {\mathrm{e}}^{tX}\left(x\right)$ is a local diffeomorphism of S.

Given $x_0\in S\subseteq {\mathbb{R}}^n$, there exists an open neighborhood $W_0$ of $x_0$ such that the restriction of X to $W_0$ extends to a vector field Y on an open subset ${\overline{U}}_0\subseteq {\mathbb{R}}^n$, containing $W_0$. We show first that the restriction of X to $W_0$ generates a local one-parameter group of local diffeomorphisms of $W_0$.

Since open sets in S are the intersections with S of open sets in ${\mathbb{R}}^n$, without loss of generality, we can write $W_0=U_0\cap S$. Let ${\mathrm{e}}^{tY}$ denote the local one-parameter group of local diffeomorphisms of $U_0$ generated by Y. There exists an open neighborhood $U_1$ of $x_0$, contained in $U_0$, and $\epsilon >0$ such that, for every $t\in (-\epsilon ,\epsilon )$, the map ${\mathrm{e}}^{tY}:U_1\to {\mathrm{e}}^{tY}\left(U_1\right)\subseteq U_0$ is a diffeomorphism of $U_1$ onto its image.

Let $W_1=U_1\cap S\subseteq W_0$. Since $Y_{\mid W_0}=X_{\mid W_0}$, the assumption that maximal integral curves of vector fields have non-empty open domains ensures that for every $x\in W_1\subseteq W_0$, there is ${\delta }_x>0$ such that ${\mathrm{e}}^{tY}\left(x\right)={\mathrm{e}}^{tX}\left(x\right)\in W_0=U_0\cap S$ for all $t\in (-{\delta }_x,{\delta }_x).$ Let ${\iota }_{W_1}=inf$$\{{\delta }_x\mid x\in W_1\}$ be the infimum of the set $\{{\delta }_x\mid x\in W_1\subseteq {\mathbb{R}}^n\}.$ Since each ${\delta }_x>0$. it follows that ${\iota }_{W_1}\ge 0$.

• If ${\iota }_{w_1}>0$, then there is a neigborhood $W_2$ of $x_0$ contained in $W_1$ and ${\epsilon }_1\in (0,\epsilon )$ such that, for every $t\in (-{\epsilon }_1,{\epsilon }_1)$, the map ${\mathrm{e}}^{tX}:W_2\to {\mathrm{e}}^{tX}\left(W_2\right)\subseteq W_1$ is a diffeomorphism of $W_2$ onto its image. In this case, the restriction of X to $W_1\ni x_0$ is a vector field on $W_1.$
• Suppose that ${\iota }_{W_1}=0.$ Since the domain of every maximal integral curve of X is open in $\mathbb{R}$, it follows that the closure ${\overline{W}}_1$ of $W_1$ has a non-empty intersection with the part of the boundary $\overline{S}\cap \left(\overline{R^n\backslash S}\right)$ of S that is not contained in S. In this case, there exists an open set $V\subset S$ such that $x_0\in V\subset \overline{V}\subset W_1$ so that $\overline{V}$ has empty intersection with the part of the boundary $\overline{S}\cap \left(\overline{R^n\backslash S}\right)$ of S that is not contained in $S.$ Then ${\iota }_V=inf\{{\delta }_x\mid x\in V\}>0$, and there exists a neighborhood $W_2$ of $x_0$ contained in V and ${\epsilon }_1\in (0,\epsilon )$ such that, for every $t\in (-{\epsilon }_1,{\epsilon }_1)$, the map ${\mathrm{e}}^{tX}:W_2\to {\mathrm{e}}^{tX}\left(W_2\right)\subseteq V$ is a diffeomorphism of $W_2$ onto its image. In this case, the restriction of X to $V\ni x_0$ is a vector field on $V.$

These arguments can be repeated for every $x_0\in W_0\subseteq S$. Hence, the restriction of X to $W_0$ is a vector field on W. Similarly, we can repeat these arguments for every $x_0\in S,$ concluding that X is a vector field on S.

Consider now the case of a general subcartesian space S. Let X be a derivation of ${\mathcal{C}}^{\infty }\left(S\right)$ such that the domains of all its maximal integral curves are open. For every $x\in S$, there exists a neighborhood W of x in S and a diffeomorphism $\chi$ of W onto a differential subspace $S_W$ of ${\mathbb{R}}^n$. Since W is open in S, maximal integral curves of the restriction $X_{\mid W}$ of X to W are open domains. The diffeomorphism $\chi :W\to S_W$ pushes forward $X_{\mid W}$ of X to a derivation ${\chi }_{*}{X}_{\mid W}$ of ${\mathcal{C}}^{\infty }\left(S_W\right)$ with the same properties. That is all integral curves of ${\chi }_{*}{X}_{\mid W}$ have open domains. By the argument above, ${\chi }_{*}{X}_{\mid W}$ is a vector field on $S_W$.

Since $\chi :W\to S_W$ is a diffeomorphism, it follows that $X_{\mid W}$ is a vector field on W. This argument can be repeated at every point $x\in S$. Therefore, for every $x\in S$, the derivation X is restricted to a vector field in an open neighborhood of x. Therefore, X is a vector field on S. □

For $X_1,\dots ,X_n\in \mathfrak{X}\left(S\right)$, consider a piecewise smooth integral curve c in $S,$ originating at $x_0\in S$, given by a sequence of steps. First, we follow the integral curve of $X_1$ through $x_0$ for time ${\tau }_1$; next, we follow the integral curve of $X_2$ through $x_1={\phi }_{{\tau }_1}^X\left(x_0\right)$ for time ${\tau }_2$, and so on. For each $i=1,\dots ,n$, let $J_i$ be $[0,{\tau }_i]\subseteq \mathbb{R}$ if ${\tau }_i>0$ or $[{\tau }_i,0]$ if ${\tau }_i<0$. Note that ${\tau }_i<0$ means that the integral curve of $X_i$ is followed in the negative time direction. For every i, $J_i$ is contained in the domain $I_{x_{i-1}}$ of the maximal integral curve of $X_i$ starting at $x_{i-1}$. In other words, for $t={\tau }_1+\dots +{\tau }_{n-1}+{\tau }_n$,

$$c\left(t\right)=c({\tau }_1+{\tau }_2+\cdots +{\tau }_{n-1}+{\tau }_n)={\phi }_{{\tau }_n}^{X_n}\circ {\phi }_{{\tau }_{n-1}}^{X_{n-1}}\circ \cdots \circ {\phi }_{{\tau }_1}^{X_1}\left(x_0\right).$$

Definition 7.

The orbit through $x_0$ of the family$\mathfrak{X}\left(S\right)$ 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 $\mathfrak{X}\left(S\right)$;

$$M=\{{\phi }_{t_n}^{X_n}\circ {\phi }_{t_{n-1}}^{X_{n-1}}\circ \cdots \circ {\phi }_{t_1}^{X_1}\left(x_0\right)\mid X_1,\dots ,X_n\in \mathfrak{X}\left(S\right)t_1,\dots ,t_n\in \mathbb{R},n\in \mathbb{N}\}.$$

Theorem 3.

Orbits M of the family $\mathfrak{X}\left(S\right)$ of vector fields on a subcartesian 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 reference [13] or the proof of Theorem 3.4.5 in [14]. □

4. Partition of $\mathit{S}$ by Orbits of $\mathfrak{X}\left(\mathit{S}\right)$

In this section, we study the consequences of Theorem 3 for our understanding of the geometry of subcartesian spaces.

Notation 1.

We denote by $\mathfrak{M}\left(S\right)$ the family of orbits of $\mathfrak{X}\left(S\right)$.

By Theorem 3, each orbit M of $\mathfrak{X}\left(S\right)$ is a manifold. Moreover, the manifold structure of $M$ is its differential structure induced by the inclusion of M in S. Hence, M is a submanifold of the differential space S. The orbits of $\mathfrak{X}\left(S\right)$ give a partition $\mathfrak{M}\left(S\right)$ of S by connected smooth manifolds. Since the notion of a vector field on a subcartesian space $S$ is intrinsically defined in terms of its differential structure, it follows that every subcartesian space has a natural partition by connected smooth manifolds. In particular, every subset S of ${\mathbb{R}}^n$ has a natural partition by connected smooth manifolds.

Proposition 2.

Let X be a derivation of $C^{\infty }\left(S\right).$ If for each $M\in \mathfrak{M}\left(S\right)$ and each $x\in M$ the maximal integral curve of X originating at $x\in M$ is contained in M, then $X\in \mathfrak{X}\left(S\right)$, that is, X, is a derivation of $C^{\infty }\left(S\right)$ that generates local one-parameter groups of local diffeomorphisms of S.

Proof. 

Suppose that X is a derivation of $C^{\infty }\left(S\right)$ satisfying the assumptions of Proposition 2. By Theorem 3, every $M\in \mathfrak{M}\left(S\right)$ is a submanifold of the differential space S. This means that the manifold structure $C^{\infty }\left(M\right)$ of M is induced by the restrictions to M of functions in $C^{\infty }\left(S\right)$. Since all integral curves of X originating at points of M are contained in M, it follows that the restriction $X_{\mid M}$ of X to M is a derivation of $C^{\infty }\left(M\right)$. However, for a manifold M, all derivations of $C^{\infty }\left(M\right)$ are vector fields on M in the sense that their integral curves generate local one-parameter groups of local diffeomorphisms of M. Moreover, domains of maximal integral curves of vector fields on a manifold are open. By assumption, this holds for every $M\in \mathfrak{M}\left(S\right)$. Since S is the union of all manifolds $M\in \mathfrak{M}\left(S\right),$ it follows that every integral curve of X has an open domain. Theorem 3 ensures that X is a vector field on S in the sense that it generates local one-parameter groups of local diffeomorphisms of S. □

Theorem 4.

The family $\mathfrak{X}\left(S\right)$ of all vector fields on a subcartesian space S is a Lie subalgebra of the Lie algebra $\mathrm{Der}{C}^{\infty }\left(S\right)$ of derivations of $C^{\infty }\left(S\right)$

This result was first obtained by Watts in his Ph.D. Thesis [17] corollary 4.71. Here, we give an alternative proof.

Proof. 

For $X\in \mathfrak{X}\left(S\right)$ and $f\in C^{\infty }\left(S\right),$ the product f$X\in \mathrm{Der}{C}^{\infty }\left(S\right)$. By construction, for every $M\in \mathfrak{M}\left(S\right)$, $X_{\mid M}$ is a vector field on the submanifold M of S, and $f_{\mid M}\in C^{\infty }\left(M\right)$. Hence, ${\left(fX\right)}_{\mid M}=f_{\mid M}{X}_{\mid M}$ is a derivation of $C^{\infty }\left(M\right)$. Therefore, for every $x\in M$, the maximal integral curve of $fX$ originating at x is the maximal integral curve of ${\left(fX\right)}_{\mid M}$ originating at x. But M is a manifold, which implies that the derivation ${\left(fX\right)}_{\mid M}$ of $C^{\infty }\left(M\right)$ is a vector field on M, so that every maximal integral curve of ${\left(fX\right)}_{\mid M}$ has an open domain.

The argument above is valid for every manifold. Since $S={\cup }_{M\in \mathfrak{M}\left(S\right)}M$, it follows that every integral curve of $fX$ has an open domain. Theorem 2 ensures that $fX$ is a vector field on S, that is $fX\in \mathfrak{X}\left(S\right)$.

Suppose that $X,Y\in \mathfrak{X}\left(S\right)$. Then, $X+Y\in \mathrm{Der}{C}^{\infty }\left(S\right)$. As before, for every $M\in \mathfrak{M}\left(S\right)$, the restrictions $X_{\mid M}$ and $Y_{\mid M}$ are vector fields on the manifold M, so that ${(X+Y)}_{\mid M}=X_{\mid M}+Y_{\mid M}$ is a vector field on M. Hence, integral curves of $X+Y$ originating at points in M have open domains. This is valid for every $M\in \mathfrak{M}\left(S\right)$, which implies that all integral curves of $X+Y$ have open domains. Therefore, $X+Y\in \mathfrak{X}\left(S\right).$

Replacing + in the arguments of the preceding paragraph by the Lie bracket $[·,·]$, we can show that, for every $X,Y\in \mathfrak{X}\left(S\right)$, their Lie bracket $[X,Y]\in \mathfrak{X}\left(S\right)$. Therefore, the family $\mathfrak{X}\left(S\right)$ of all vector fields on S is a Lie subalgebra of $\mathrm{Der}{C}^{\infty }\left(S\right)$. □

Proposition 3.

(Frontier Condition) For $M,M^{\prime }\in \mathfrak{M}\left(S\right),$ if $M^{\prime }\cap \overline{M}\ne \varnothing$, then either $M^{\prime }=M$ or $M^{\prime }\subset \overline{M}\backslash M$.

Proof. 

Let M and $M^{\prime }$ be orbits of $\mathfrak{X}\left(S\right)$ such that $M^{\prime }\cap \overline{M}\ne \varnothing$, where $\overline{M}$ denotes the closure of M in S. Suppose that $x_0\in M^{\prime }\cap \overline{M}$ with $M^{\prime }\ne M$. Let ${\left\{x_k\right\}}_{k\in \mathbb{N}}$ be a sequence of points in M converging to $x_0$. For every $X\in \mathfrak{X}\left(S\right)$, there is an open neighborhood $U_0$ of $x_0$ in S and $t_0>0$ such that $exp\left(tX\right)\left(x\right)$ is defined for every $0\le t\le t_0$ and every $x\in U_0$. Moreover, if $0\le t\le t_0$, the map $U_0\to S:x\mapsto exp\left(tX\right)\left(x\right)$ is continuous. Therefore, for $0\le t\le t_0$,

$$\underset{k\to \infty }{lim}exp\left(tX\right)\left(x_k\right)=exp\left(tX\right)\left(x_0\right).$$

Since M is the orbit of $\mathfrak{X}\left(S\right)$, it is invariant under the family of one-parameter local groups of local diffeomorphisms of S generated by vector fields, and ${\left\{x_k\right\}}_{k\in \mathbb{N}}\subseteq M$, it follows that ${lim}_{k\to \infty }exp\left(tX\right)\left(x_k\right)\in \overline{M}$. Therefore, $exp\left(tX\right)\left(x_0\right)\in \overline{M}$. On the other hand, $M^{\prime }$ is the orbit of $\mathfrak{X}\left(S\right)$ through $x_0$, so that $exp\left(tX\right)\left(x_0\right)\in M^{\prime }$. Hence, $exp\left(tX\right)\left(x_0\right)\in M^{\prime }\cap \overline{M}$. By assumption, $M^{\prime }\ne M,$ which implies that $exp\left(tX\right)\left(x_0\right)\in \overline{M}\backslash M$. This holds for every $X\in$$\mathfrak{X}\left(S\right)$ and $x_0\in M^{\prime }\cap$$\overline{M}\backslash M$. Therefore, $M^{\prime }\subset \overline{M}\backslash M$. □

Proposition 4.

(Whitney’s Conditions A and B) Consider a differential subspace S of ${\mathbb{R}}^n$. Let $y\in M^{\prime }\subseteq \overline{M}\backslash M,$ where $M,M^{\prime }\in \mathfrak{M}\left(S\right)$, and let $m=dimM$.

A. 

If $x_i$ is a sequence of points in M such that $x_i\to y\in M^{\prime }$, and $T_{x_i}M$ converges to some m-plane $E\subseteq$$T_{y}S\subseteq T_y{\mathbb{R}}^n$ then $T_y{M}^{\prime }\subseteq E$.

B. 

If $y_i$ is a sequence of points in $M^{\prime }$ also converging to y, suppose that $T_{x_i}M$ converges to an m-plane $E\subseteq$$T_{y}S\subseteq T_y{\mathbb{R}}^n$ and the secant $\overleftrightarrow{x_i{y}_i}$ converges to some line in $L\subseteq {\mathbb{R}}^n$. Then $L\subseteq E.$

Proof. 

A. Since M is a submanifold of the differential subspace S of ${\mathbb{R}}^n$, and $\overline{M}$ is the closure of M in S, then $\overline{M}$ is a differential subspace of S. Moreover, M and $M^{\prime }$ are submanifolds of $\overline{M}$. Hence, for a sequence $x_i$ in M, such that $y={lim}_{i\to \infty }{x}_i\in M^{\prime }$, we have

$$T_y\overline{M}=\underset{i\to \infty }{lim}{T}_{x_i}M.$$

Since $M^{\prime }$ is a submanifold of $\overline{M}$, it follows that $T_y{M}^{\prime }\subseteq T_y\overline{M}$.

In order to write the result in the form used in the statement of the proposition, we use the identification ${\mathbb{R}}^n\times {\mathbb{R}}^n\equiv T{\mathbb{R}}^n$ such that the following diagram commutes

$$\begin{array}{ccccc}& {\mathbb{R}}^n\times {\mathbb{R}}^n& \equiv & T{\mathbb{R}}^n& \\ {\mathrm{pr}}_1& \downarrow & & \downarrow & \tau \\ & {\mathbb{R}}^n& =& {\mathbb{R}}^n\end{array},$$

where ${\mathrm{pr}}_1:{\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^n$ is the projection on the first factor, and $\tau :T{\mathbb{R}}^n\to {\mathbb{R}}^n$ is the tangent bundle projection. Moreover, for every $f\in C^{\infty }\left({\mathbb{R}}^n\right)$ and $v=(\mathit{x},\mathit{v})\in T{\mathbb{R}}^n$, the derivation of f by v is $vf=〈df\mid \mathit{v}〉\left(\mathit{x}\right)$. With this identification, the m-plane $T_y\overline{M}\subseteq$$T_{\mathit{y}}S\subseteq T_{\mathit{y}}{\mathbb{R}}^n$, can be expressed as $T_y\overline{M}=(y$,$E)$, where $E\subseteq {\mathbb{R}}^n$. Hence, $T_y{M}^{\prime }\subseteq (y,E).$

B. The sequence $\overleftrightarrow{x_i{y}_i}$ of secants, if it converges as $i\to \infty ,$ defines a derivation $v\in T_y\overline{M}$ such that, for every $f\in C^{\infty }\left(\overline{M}\right)$,

$$vf=\underset{i\to \infty }{lim}\frac{f\left(x_i\right)-f\left(y_i\right)}{∥x_i-y_i∥},$$

where $∥x_i-y_i∥={\left({\sum }_{i=1}^n{(x_i-y_i)}^2\right)}^{1/2}$. The limiting line of the sequence $\overleftrightarrow{x_i{y}_i}$ of secants is the line L through y in direction v. Since $v\in T_y\overline{M}$, in the identification used above, $L\subseteq E$. □

For each $n=0,1,2,\dots ,$ let

$${\mathfrak{M}}_n\left(S\right)=\{M\in \mathfrak{M}\left(S\right)\mid dimM=n\},$$

(8)

and

$$S_n=\coprod _{M\in {\mathfrak{M}}_n\left(S\right)}M.$$

(9)

Since elements of ${\mathfrak{M}}_n\left(S\right)$ are mutually disjoint n-dimensional manifolds, it follows that $S_n$ is a manifold of dimension n, and the connected manifolds $M\in {\mathfrak{M}}_n\left(S\right)$ are connected components of $S_n.$ Since S is a subcartesian space, the dimension n of $S_n$ is locally bounded. For every chart $\alpha :V_{\alpha }\to W_{\alpha }\subseteq {\mathbb{R}}^{d_{\alpha }}$, $dim{S}_n\cap V_{\alpha }\le d_{\alpha }$. Hence,

$$S=\coprod _{n=0}^{\infty }{S}_n.$$

(10)

In general, the partition $\mathfrak{M}\left(S\right)$ of S by orbits of $\mathfrak{X}\left(S\right)$ need not be locally finite, as is shown in the following example.

Example 1.

Let $S=\mathbb{Q}\times \mathbb{R}\subseteq {\mathbb{R}}^2$, where $\mathbb{Q}$ is the set of rational numbers. A derivation $X\in \mathrm{Der}S$ is a vector field only if it is tangent to the second factor $\mathbb{R}$. In other words, if $f\in C^{\infty }\left(S\right)$ is written in terms of the coordinates $(x_1,x_2)\in \mathbb{Q}\times \mathbb{R}$, then $X\in \mathfrak{X}\left(S\right)$ if, and only if, there exists $a\in C^{\infty }\left(S\right)$ such that

$$\left(Xf\right)(x_1,x_2)=a(x_1,x_2)\frac{\partial f(x_1,x_2)}{\partial x_2}$$

for every $f\in C^{\infty }\left(S\right)$ and every $(x_1,x_2)\in \mathbb{Q}\times \mathbb{R}$.

Since the space $\mathfrak{X}\left(\mathbb{R}\right)$ of vector fields on $\mathbb{R}$ acts transitively on $\mathbb{R}$, it follows that in our example, for every $x=(x_1,x_2)\in S$, the orbit M of $\mathfrak{X}\left(S\right)$ through $x=(x_1,x_2)$ is $\left\{x_1\right\}\times \mathbb{R}$. Thus, the space $\mathfrak{M}\left(S\right)$ of orbits of $\mathfrak{X}\left(S\right)$ for $S=\mathbb{Q}\times \mathbb{R}$ is parametrized by $\mathbb{Q},$ and it is not locally finite.

Example 2.

Let $S=\{x\in \mathbb{R}\mid x=0$ or $x=\frac{1}{n}$ for $n\in \mathbb{N}\}$. In this case, the only vector field on S is $X=0$, and every $M\in \mathfrak{M}\left(S\right)$ is a single point. There is no neighborhood of $0\in S$ that contains only a finite number of points of S. Hence, $\mathfrak{M}\left(S\right)$ is not locally finite.

5. Comparison with Stratification

There are several definitions of stratification of a closed subset S of a smooth manifold. Here, we only consider the $C^{\infty }$ category. The definition used by Goresky and MacPherson [18], adapted to the setup considered here, can be reformulated as follows.

Definition 8.

A partition of a subcartesian space S by submanifolds of S is a decomposition of S if it is locally finite and satisfies the frontier condition, which is the statement of Proposition 3. A Whitney stratification of S is a decomposition of S that satisfies Whitney’s conditions A and B, which is the statement of Proposition 4.

If S is a closed subset of a smooth manifold M, then by composing the inclusion of S into M with the charts for M, we obtain an atlas $\mathfrak{A}\left(S\right)=\{\alpha :V_{\alpha }\to W_{\alpha }\},$ where $V_{\alpha }$ an open subset of S and $W_{\alpha }$ is a locally closed subset of ${\mathbb{R}}^{d_{\alpha }}.$ In other words, S is a locally closed subcartesian space. Propositions 3 and 4 ensure that if S is a locally closed subcartesian space and the partition $\mathfrak{M}\left(S\right)$ is locally finite, then $\mathfrak{M}\left(S\right)$ is a Whitney stratification of S.

Mather [19] uses the term prestratification for a decomposition of S by submanifolds and the term stratification for the sheaf $\mathcal{S}$ of germs of manifolds of prestratification. If S is locally closed and $\mathfrak{M}\left(S\right)$ is locally closed, then $\mathfrak{M}\left(S\right)$ is a prestratification of S and the sheaf $\mathcal{S}$ of germs of manifolds in $\mathfrak{M}\left(S\right)$ is the induced stratification.

Prestratifications of S that induce the same sheaf of germs $\mathcal{S}$ can be partially ordered by inclusion. Pflaum [20], identifies the sheaf $\mathcal{S}$ of germs of the manifolds of prestratification with the coarsest prestratification in this class. If S is locally closed and $\mathfrak{M}\left(S\right)$ is locally closed, then the coarsest prestratification in the sense of Pflaum is ${\left\{S_n\right\}}_{n=0}^{\infty }$, where $S_n={\coprod }_{M\in {\mathfrak{M}}_n\left(S\right)}M$ – see Equation (9).

We have seen that, for every definition of stratification discussed above, if S is a locally closed subcartesian space and $\mathfrak{M}\left(S\right)$ is locally finite, then the decomposition $\mathfrak{M}\left(S\right)$ of S corresponds to a stratification of S. It should be noted that, in this case, our approach corresponds to an algorithm leading to the discovery of the stratification of S. Once S is chosen and its differential structure is established, there is no room for choice. The main step is to determine the family $\mathfrak{X}\left(S\right)$, consisting of all derivations of $C^{\infty }\left(S\right)$ that generate local one-parameter groups of local diffeomorphisms of S. Theorem 2 helps us make this determination.

6. Transient Derivations

Up to now, we have concentrated on orbits of the Lie algebra $\mathfrak{X}\left(S\right)$ of vector fields on S; that is, derivations of $C^{\infty }\left(S\right)$ that generate local one-parameter local groups of diffeomorphisms. In this section, we consider the role played by the derivations of $C^{\infty }\left(S\right)$ that do not generate local one-parameter groups of local diffeomorphisms of S.

Definition 9.

Transient derivation on a subcartesian space S is a derivation of $C^{\infty }\left(S\right)$ that does not generate local one-parameter groups of local diffeomorphisms of S

The term transient derivation is an extension of the notion of transient vector field used in the theory of manifolds with a boundary [15].

Let X be a transient derivation on a subcartesian space S. By Theorem 1, for every $x_0\in S$, there exists a unique maximal integral curve $c_0$ of X such that $c_0\left(0\right)=x_0$. If, for every $x_0\in S$, the maximal integral curve $c_0$ of X through $x_0\in M\in \mathfrak{M}\left(S\right)$ is contained in $M_0$, then Proposition 2 ensures that X generates one-parameter local groups of diffeomorphisms of S, which contradicts the assumption that X is a transient derivation. Therefore, there must exist a maximal integral curve $c:I\to S$ of X such that, for some $t_1\in I$, the curve c crosses from a manifold $M\in \mathfrak{M}\left(S\right)$ to a manifold $M^{\prime }\subseteq \overline{M}\backslash M$. It follows that transient derivations provide integral curves joining manifolds of $\mathfrak{M}\left(S\right)$.

7. Manifolds with Corners

Manifolds with corners are a basic example of stratified subcartesian spaces. Here, we rely on the presentation of the theory of manifolds with corners given in Joyce [16]. We begin with a definition of a manifold with corners as a local Euclidean Hausdorff manifold—see Definition 2.6. This definition is equivalent to the original definition by Cerf [2] used in [16].

Definition 10.

A d-dimensional manifold with corners is a paracompact Hausdorff topological space S equipped with a maximal d-dimensional atlas $\mathfrak{A}=\{\alpha :V_{\alpha }\to W_{\alpha }\}$, where α is a homeomorphism of an open subset $V_{\alpha }$ of S onto an open subset $W_{\alpha }$ of ${\mathbb{R}}_{k_{\alpha }}^d={[0,\infty )}^{k_{\phi }}\times {\mathbb{R}}^{d-k_{\alpha }}\subseteq {\mathbb{R}}^d$, in the topology induced by its inclusion in ${\mathbb{R}}^d,$ which satisfies the conditions listed below.

1.

The sets$\{V_{\phi }\mid \alpha \in \mathfrak{A}\}$ form a covering of S.

2.

For every $\alpha ,\beta \in \mathfrak{A}$,and every $x\in V_{\alpha }\cap V_{\beta },$ there exist:

(a) 

a $C^{\infty }$-mapping ${\mathsf{\Phi }}_{\alpha }$ of an open neighborhood $U_{\alpha }$ of $\alpha \left(x\right)\in {\mathbb{R}}^{n_{\alpha }}$ to ${\mathbb{R}}^{n_{\beta }},$ which extends the mapping

$$\beta \circ {\alpha }^{-1}:\alpha (V_{\alpha }\cap V_{\beta })\to \beta (V_{\alpha }\cap V_{\beta }),$$

(b) 

a $C^{\infty }$-mapping ${\mathsf{\Phi }}_{\beta }$ of an open neighborhood $U_{\beta }$ of $\beta \left(x\right)\in {\mathbb{R}}^{n_{\beta }}$ to ${\mathbb{R}}^{n_{\alpha }},$ which extends the mapping

$$\alpha \circ {\beta }^{-1}:\beta (V_{\alpha }\cap V_{\beta })\to \alpha (V_{\alpha }\cap V_{\beta }).$$

3.

A continuous function $f:S\to \mathbb{R}$ on S is smooth if and only if, for every chart $\alpha :V_{\alpha }\to W_{\alpha }\subseteq {\mathbb{R}}^d$, there exists an open set $U_{\alpha }$ in ${\mathbb{R}}^d$ containing $W_{\alpha }$, and a smooth function $F\in C^{\infty }\left(U_{\alpha }\right)$ such that $f\circ {\alpha }^{-1}:W_{\alpha }\to \mathbb{R}$ is the restriction of F to $W_{\alpha }\subseteq U_{\alpha }$. We denote by $C^{\infty }\left(S\right)$ the space of smooth functions on S.

4.

A map $\phi :S\to$R between manifolds with corners S and R is smooth if it is continuous and, for every pair of charts $\alpha :V_{\alpha }\to W_{\alpha }\subseteq {\mathbb{R}}^{d_S}$ in $\mathfrak{A}\left(S\right)$ and $\beta :V_{\beta }\to W_{\beta }\subseteq {\mathbb{R}}^{d_R}$ in $\mathfrak{A}\left(R\right),$ such that $\phi \circ {\alpha }^{-1}\left(W_{\alpha }\right)\subseteq V_{\beta },$ there exist open subsets $U_{\alpha }\subseteq {\mathbb{R}}^{d_S},$ $U_{\beta }\subseteq {\mathbb{R}}^{d_R}$ and $F_{\alpha \beta }\in C^{\infty }(U_{\alpha },U_{\beta })$ such that: (i) $W_{\alpha }\subseteq U_{\alpha },$ (ii) $W_{\alpha }\subseteq U_{\alpha }$ and, for every $\mathit{x}\in W_{\alpha },$

$$F_{\alpha \beta \mid W_a}\left(\mathit{x}\right)=\beta \circ \phi \circ {\alpha }^{-1}\left(\mathit{x}\right).$$

The fundamental notion of a manifold with corners S, leading to the stratification structure of S, is the depth functions

$$\mathrm{depth}{ }_S:S\to {\mathbb{Z}}_{\ge 0}:x\mapsto \mathrm{depth}{ }_{S}x=\underset{\alpha \in \mathfrak{A}}{min}\{k_{\alpha }\mid x\in V_{\alpha }\}.$$

It is easy to show that the function $\mathrm{depth}{ }_{S}x$ is well defined by the differential structure $C^{\infty }\left(S\right)$ of the manifold with corners S under consideration.

Definition 11.

For each $k\ge 0,$ the depth k stratum of S is

$$S^k=\{x\in S\mid {\mathrm{depth}}_S x=k\}.$$

Proposition 5.

Let S be a d-dimensional manifold with corners.

(a) 

S is a disjoint union of $S^k,$ for $k=0,\dots ,d$, that is, $S={\coprod }_{k=0}^d{S}^k$.

(b) 

Each $S^k$ has the structure of a $(d-k)$-dimensional manifold (without boundaries or corners).

(c) 

If $\overline{S^k}\cap S^l\ne \varnothing$, then either $S^l=S^k$, or $S^l\subseteq \overline{S^k}\backslash S^k,$ where $\overline{S^k}$ denotes the closure of $S^k$ in S.

(d) 

For every $k=0,\dots ,d,$

$$\overline{S^k}=\coprod _{l=k}^d{S}^l$$

(11)

is a manifold with corners.

Proof. 

(a) The depth of $x\in S$ is uniquely defined by the maximal n-dimensional atlas $\mathfrak{A}.$ Hence, $S^k\cap S^l=\varnothing$ if $k\ne l$. Moreover, $k=0,\dots ,d$. Hence, S is a disjoint union of $S^k,$ for $k=0,\dots ,d.$

(b) Definition 1 ensures that S has an atlas $\mathfrak{A}=\{\alpha :V_{\alpha }\to W_{\alpha }\}$, where $\alpha$ is a homeomorphism of an open subset $V_{\alpha }$ of S onto an open subset $W_{\alpha }$ of ${\mathbb{R}}_{k_{\alpha }}^d={[0,\infty )}^{k_{\phi }}\times {\mathbb{R}}^{d-k_{\alpha }}\subseteq {\mathbb{R}}^d$, in the topology induced by its inclusion in ${\mathbb{R}}^d.$ For each $x\in S^k\subseteq S$, there exists a chart $\alpha :V_{\alpha }\to W_{\alpha }$ for S such that $x\in V_{\alpha }$, and $W_{\alpha }=$$({[0,\infty )}^k\times {\mathbb{R}}^{d-k})\cap U_{\alpha }$, where $U_{\alpha }$ is an open subset of ${\mathbb{R}}^d$. Moreover, $\alpha (V_{\alpha }\cap S^k)=({\left[0\right]}^k\times {\mathbb{R}}^{d-k})\cap U_{\phi }$. Note that ${\left[0\right]}^k\times {\mathbb{R}}^{d-k}\cong {\mathbb{R}}^{d-k}$ and ${\mathbb{R}}^{d-k}\cap U_{\phi }$ is an open subset of ${\mathbb{R}}^{d-k}$. The collection of charts

$$\begin{array}{ccc}\hfill {\mathfrak{A}}_{S^k}& =& \{{\phi }_{\mid V_{\phi }\cap S^k}:V_{\phi }\cap S^k\to \phi (V_{\phi }\cap S^k){\mathbb{R}}^{d-k}\cap U_{\phi }\mid \hfill \\ & & \mathrm{for}\mathrm{all}\phi \in \mathfrak{A}\mathrm{such}\mathrm{that}\phi (V_{\phi }\cap S^k)=({\left[0\right]}^k\times {\mathbb{R}}^{d-k})\cap U_{\phi }\}\hfill \end{array}$$

is a $(d-k)$-manifold atlas for $S^k$. It satisfies the condition (2) of Definition 1 because the atlas $\mathfrak{A}$ satisfies this condition.

(c) Recall that a manifold with corners S is defined as a topological space satisfying certain conditions. Therefore, by the closure $\overline{S^k}$ of $S^k$, we mean the closure of $S^k$ in S. If S were a subset of some other topological space $T,$ then the closure of $S^k$ in S is the intersection with S of the closure of $S^k$ in the topology induced by its embedding of S into T.

If $\overline{S^k}\cap S^l\ne \varnothing$, there exists $x_0\in \overline{S^k}\cap S^l$$\subseteq S$. Since $x_0\in \overline{S^k}$, every open neighborhood V of x has a non-empty intersection with $S^k$. Since $x_0\in S^l$, it follows that depth $x_0=l$, and there exists a chart $\alpha :V_{\alpha }\to W_{\alpha }$ such that $x_0\in V_{\alpha }$, and $W_{\alpha }=$$({[0,\infty )}^l\times {\mathbb{R}}^{d-l})\cap U_{\alpha }$, where $U_{\alpha }$ is an open subset of ${\mathbb{R}}^{d-l}.$ Without loss of generality, we may assume that, for each $x\in S^k\cap V_{\alpha }$, $\alpha \left(x\right)=(x^1,\dots ,x^l,x^{l+1},\dots ,x^d),$ has the first k of the l components $(x^1,\dots ,x^l)$ equal to zero. Hence, $l\ge k$. If $l=k$, then $S^l=S^k$. If $l>k$, then $x_0\in \overline{S^k}\backslash S^k.$ This argument holds for every $x\in \overline{S^k}\cap S^l$ with $l>k$. Hence, $S^l\subseteq \overline{S^k}\backslash S^k$.

(d) It follows from (a) and (c) that

$$\overline{S^k}=\overline{S^k}\cap S=\overline{S^k}\cap \coprod _{l=0}^d{S}^l=\coprod _{l=0}^d\overline{S^k}\cap S^l=\coprod _{l=k}^d\overline{S^k}\cap S^l=\coprod _{l=k}^d{S}^l.$$

It is easy to check that $\overline{S^k}={\coprod }_{l=k}^d{S}^l$ satisfies the conditions for a manifold with corners. □

Definition 11 quotes the corresponding definition in [16], in which the term “depth k stratum” is used without explanation. It shows that the stratification structure of manifolds with corners is common knowledge in this field. By Definition 10, manifolds with corners are locally closed subcartesian spaces.

All definitions of stratifications discussed in the preceding section deal with closed subsets of a manifold. Every closed subset of a manifold is a locally closed subcartesian space. However, not every locally closed subcartesian space can be presented as a closed subset of a manifold. Hence, the use of the term “stratification” in the theory of manifolds with corners is a generalization of the classical notion of stratification, which is convenient to adopt in the theory of differential spaces.

In order to relate the general theory of the preceding sections to the example of manifolds with corners, we have to establish what the vector fields are on manifolds with corners. In other words, we have to establish the class of derivations of $C^{\infty }\left(S\right)$ that generate local one-parameter groups of local diffeomorphisms of S.

The depth function stratification $\{S^0,S^1,\dots ,S^k,\dots ,S^d\}$ encodes the intrinsic geometric structure of the manifold with corners S. Therefore, we may expect that connected components of the strata of the stratification $\{S^0,S^1,\dots ,S^k,\dots ,S^d\}$ are integral manifolds of the Lie algebra $\mathfrak{X}\left(S\right)$ of S. We establish this result in a series of propositions.

Proposition 6.

Let S be a manifold with corners. A derivation X of $C^{\infty }\left(S\right)$ is a vector field on S if and only if every maximal integral curve $c:$$I\to S$ of X is contained in a single stratum of the depth function stratification of S.

Proof. 

Let X be a derivation of $C^{\infty }\left(S\right)$ of a d-manifold with corners. Suppose that every maximal integral curve $c:$$I\to S$ of X is contained in a single stratum in $\mathfrak{M}\left(S\right).$ Let M be a connected component of a stratum $S^k$ of the depth function stratification of S. Since all integral curves of X are connected, it follows that all integral curves of X originating at points in M are contained in $M.$ Therefore, the restriction $X_{\mid M}$ of X to M is a derivation of $C^{\infty }\left(M\right)$. However, M is a manifold, and all derivations of $C^{\infty }\left(M\right)$ are vector fields on M. Therefore, $X_{\mid M}$ generates a local one-parameter group of local diffeomorphisms of M.

The argument above is valid for every connected component of each stratum of the depth function stratification of S. Therefore, the derivation X generates a local one-parameter group of local diffeomorphisms of manifolds with corners that preserve the depth function stratification of S. Hence, the derivation X is a vector field on S.

Let X be a vector field on $S.$ That is, X generates a local one-parameter group of local diffeomorphisms of X. We need to show that every integral curve of X is contained in a connected component of a single stratum of the depth function stratification of S. We suppose the opposite and derive a contradiction.

Suppose that there is an integral curve $c:$ $I\to S$ of X such that, for $-ϵ<$$t<0$, $c\left(t\right)$ is in a connected component M of a stratum $S^m$ and $c\left(0\right)$ is in a connected component N of a different stratum $S^n$ of S. Since $c\left(0\right)={lim}_{t\to 0^{-}}c\left(t\right)$, Proposition 5(c) implies that $N\subseteq \overline{M}\backslash M$ so that $m\le n-1$. Let $\alpha :V\to W\subseteq {\mathbb{R}}^d$ be a chart in $\mathfrak{A}$, where V is a neighborhood of $c\left(0\right)$ in S and $W\subseteq$${\mathbb{R}}_m^d\cap U=({[0,\infty )}^m\times {\mathbb{R}}^{d-m})\cap U\subseteq {\mathbb{R}}^d$ for some open neighborhood U of ${\mathbf{0}}_d\in {\mathbb{R}}^d,$ such that $\alpha \left(c\left(0\right)\right)={\mathbf{0}}_d\in {\mathbb{R}}^d$. Moreover, $\alpha (M\cap V)=(\left\{{\mathbf{0}}_m\right\}\times {\mathbb{R}}^{d-m})\cap U$, and for every

$$\mathit{x}=(x^1,\dots ,x^m,x^{m+1},\dots ,x^n,x^{n+1},\dots ,x^d)\in \alpha (M\cap V),$$

the first m coordinates $(x^1,\dots ,x^m)$ are equal to zero. Similarly, $\alpha (N\cap V)=(\left\{{\mathbf{0}}_n\right\}\times {\mathbb{R}}^{d-n})\cap U$, and for every $\mathit{y}=(y^1,\dots ,y^m,y^{m+1},\dots ,y^n,x^{n+1},\dots ,x^d)\in \alpha (N\cap V),$ the first n coordinates $(y^1,\dots ,y^m,y^{m+1},\dots ,y^n)$ are equal to zero.

For every $t<0$, there exists a neighborhood $V_t$ of $c\left(t\right)$ in V such that $V_t\cap N=\varnothing$. Therefore, there exists an open neighborhood $U_t$ of ${\mathbf{0}}_d\in {\mathbb{R}}^d$ such that

$$\alpha \left(V_t\right)=({[0,\infty )}^m\times {\mathbb{R}}^{d-m})\cap U_t.$$

On the other hand, if $V_0\subseteq V$ is a neighborhood of $c\left(0\right)$ in S, then

$$\alpha \left(V_0\right)=({[0,\infty )}^n\times {\mathbb{R}}^{d-n})\cap U_0$$

for a neighborhood $U_0$ of ${\mathbf{0}}_d\in {\mathbb{R}}^d$. However, $m\ne n$, so that, for $t<0$, $\alpha \left(V_t\right)$ is not diffeomorphic to $\alpha \left(V_0\right)$. Since $\alpha :V\to W$ is a diffeomorphism, it follows that $V_t$ is not diffeomorphic to $V_0$ for every $t<0$. This contradicts the assumption that X generates a local one-parameter group of local diffeomorphisms of S. □

Proposition 7.

Let S be a manifold with corners and X a derivation on S such that, for every connected component M of the depth function stratification of S, the restriction $X_{\mid M}$ of X to M is a vector field on the manifold M. Then, X is a vector field on S.

Proof. 

In view of Proposition 6, it suffices to show that every integral curve of X originating at a connected component M of the depth function stratification of S is contained in M. Suppose that there is an integral curve $c:I\to S$ of $X,$ originating at $x_0$ in a connected component M of a stratum of the depth function stratification of S, such that $x_1=c\left(t_1\right)\in N\in \overline{M}\backslash M$, where $t_1=min\{t\in I\mid t>0$ and $c\left(t\right)\in M\}$ and N is a connected component of another stratum in S. Since X is of class $C^{\infty }$, it follows that

$$\underset{t\to t_1-}{lim}X\left(c\left(t\right)\right)=\underset{t\to t_{1-}}{lim}{X}_{\mid M}\left(c\left(t\right)\right)=X\left(c\left(t_1\right)\right)=X_{\mid N}\left(c\left(t_1\right)\right).$$

Suppose that $X\left(x_1\right)=0$. The equation

$$\frac{df}{dt}\left(c\left(t\right)\right)=\left(Xf\right)\left(c\left(t\right)\right)$$

for every $f\in C^{\infty }\left(S\right)$ implies that,

$$\frac{dt}{df}\left(c\left(t\right)\right)=\frac{1}{\left(Xf\right)\left(c\right(t\left)\right)}.$$

Hence, $X\left(x_1\right)=0$ implies that $t\to \infty$ as $c\left(t\right)\to x_1.$ Therefore, $x_1={lim}_{t\to \infty }c\left(t\right)$, and it is not in the range of the curve c, contrary to the previous assumption.

Suppose now that $X\left(x_1\right)=X_{\mid N}\left(x_1\right)\ne 0$. Note that $X_{\mid N}$ is a vector field on the manifold N. Hence, there exists an integral curve $c_N:I_N\to N$ of $X_{\mid N}$ originating at $x_1$. Consider a chart $\alpha :V\to W$ in $\mathfrak{B}$ such that V isa neighborhood of $x_1=c\left(t_1\right),$ and $W\subseteq {\mathbb{R}}_n^d={[0,\infty )}^n\times {\mathbb{R}}^{d-n}\subseteq {\mathbb{R}}^d$ contains $\alpha \left(x_1\right)$. By Proposition 3.1.6 in [14], there exists a neighborhood $V_1$ of $x_1\in V,$$U\subseteq {\mathbb{R}}^d$ such that ${\alpha }_{\mid V_1}:V_1\to W_1=\alpha \left(V_1\right)\subseteq W$ is a diffeomorphism, and a vector field Y defined on an open set $U\subseteq {\mathbb{R}}^d$ containing $W_1$ such that ${\left({\alpha }_{*}X\right)}_{\mid W_1}=Y_{\mid W_1}$.

Since $c_N:I_N\to N$ of $X_{\mid N}$ originates at $x_1\in V_1$, it follows that there is a connected subset ${\tilde{I}}_N$ of $I_N$ containing 0 such that the restriction ${\tilde{c}}_N$ of $(\alpha \circ c_N)$ to ${\tilde{I}}_N$ has its range in $W_1$. The equation above implies that ${\tilde{c}}_N:{\tilde{I}}_N\to W_1$ is an integral curve of Y originating at $\alpha \left(x_1\right)$. On the other hand, $x_1=c\left(t_1\right)$. Hence, $c^{\prime }:I^{\prime }\to S:t\mapsto c(t-t_1)$ is an integral curve of X originating at $x_1=c\left(t_1\right)$, where $I^{\prime }$ is I shitfed by $t_1$. Let ${\tilde{I}}^{\prime }$ be a connected neighborhood of $0\in I^{\prime }$ such that the restriction ${\tilde{c}}^{\prime }$ of $\alpha \circ c^{\prime }$ to ${\tilde{I}}^{\prime }$ has its range in $W_1$. As before, ${\tilde{c}}^{\prime }:{\tilde{I}}^{\prime }\to W_1$ is an integral curve of Y originating at $\alpha \left(x_1\right)$. However, Y is a vector field on an open subset of ${\mathbb{R}}^d,$ and the germ of its integral curve passing through $\alpha \left(x_1\right)$ is unique up to parametrization. However, ${\tilde{c}}_N$ and ${\tilde{c}}^{\prime }$ are distinct integral curves of Y such that ${\tilde{c}}_N\left(0\right)=$ ${\tilde{c}}^{\prime }\left(0\right)=\alpha \left(x_1\right)$. Therefore, we have a contradiction with the hypothesis that $X\left(x_1\right)\ne 0.$ □

Proposition 8.

Let S be a d-manifold with corners. For every vector ${\mathrm{v}}_0\in T_{x_0}S$ tangent to the stratum of the depth function stratification of S that contains $x_0=\tau \left({\mathrm{v}}_0\right)$, there exists a vector field X on S extending ${\mathrm{v}}_0$; that is, $X\left(x_0\right)={\mathrm{v}}_0.$

Proof. 

If ${\mathrm{v}}_0=0$, then it extends to the vector field $X=0$ on S. That is, $Xf=0$ for every $f\in C^{\infty }\left(S\right)$.

If ${\mathrm{v}}_0\ne 0$, consider a chart $\alpha :V_{\alpha }\to W_{\alpha }\subseteq {\mathbb{R}}_n^d=$${[0,\infty )}^n\times {\mathbb{R}}^{d-n}\subseteq {\mathbb{R}}^d$ on the manifold with corners S such that $V_{\alpha }$ is a neighborhood of x in S and ${\mathbb{R}}^d.$ If $\mathrm{depth}{ }_{S}x=n$ then, without loss of generality, we may assume that

$$\begin{array}{ccc}\hfill \alpha \left(x_0\right)& =& {\mathit{x}}_0=(x_0^1,\dots ,x_0^d),\mathrm{where}{x}_0^1=\dots =x_0^n=0\mathrm{and}{x}_0^{n+1}=\dots =x_0^d=1,\hfill \\ \hfill \alpha (V_{\alpha }\cap S^n)& =& \{(x^1,\dots ,x^d)\in {\mathbb{R}}^d\mid x^1=0,\dots ,x^n=0,(x^{n+1} ,\dots ,x^d)\in U\subseteq {\mathbb{R}}^{d-n}\}\hfill \\ & =& \left\{{\mathbf{0}}_n\right\}\times U\subset {\mathbb{R}}^d,\hfill \end{array}$$

where U is open in ${\mathbb{R}}^{d-n}$. For every $m\in 0,1,\dots ,d,$ the point $(x^1,\dots ,x^d)\in \alpha (V_{\alpha }\cap S^m)$ if and only if exactly m of the coordinates $x^1,\dots ,x^d$ are zero. A vector $\mathit{v}=(v^1,\dots ,v^d)$ is tangent to $\alpha (V_{\alpha }\cap S^m)$ at $(x^1,\dots ,x^d)\in \alpha (V_{\alpha }\cap S^m)$ if and only if, for every $i=1,\dots ,d,$$x^i=0$ implies $v^i=0$. Since $\alpha :V_{\alpha }\to W_{\alpha }\subseteq R^d$ is a diffeomorphism, and the definition of the depth function is independent of the chart, it follows that $v\in T_{x}S$ is tangent at x to $S^m$ if and only if $\mathit{v}=T\alpha \left(v\right)$ is tangent to $\alpha (V_{\alpha }\cap S^m)$ at the point $\alpha \left(x\right)=(x^1,\dots ,x^d)$.

Thus, for $x\in S^m\cap V_{\alpha }$ a vector $\mathrm{v}\in T_{x}S$ is in $T_x{S}^m$ if and only if $x^i{v}^i=0$ for every $i=1,\dots ,d$, where $(x^1,\dots ,x^d)$ are coordinates of $\alpha \left(x\right)$ in ${\mathbb{R}}^d$ and $(v^1,\dots ,v^d)$ are components of $T\alpha \left(\mathrm{v}\right)\in T_{\alpha \left(x\right)}$ ${\mathbb{R}}^d\cong T_{\alpha \left(x_0\right)}{\mathbb{R}}^d$.

Since U is open in ${\mathbb{R}}^{d-n}$, there exists $ϵ\in (0,\frac{1}{2})$ such that the set

$$\begin{array}{ccc}\hfill W{ }_{ϵ}=\{\mathit{x}=(x^1,\dots ,x^d)\in {\mathbb{R}}^d& \mid & -ϵ<x^i<ϵ\mathrm{for}i=1,\dots ,n\hfill \\ \hfill \mathrm{and}1-ϵ& <& x^j<1+ϵ\mathrm{for}j=n+1,\dots ,d\}\hfill \end{array}$$

is an open neighborhood of $\alpha \left(x_0\right)={\mathit{x}}_0$ in $W_{\alpha }\subseteq {\mathbb{R}}^d$ and ${\overline{W }}_{ϵ}\subseteq$$W_{\alpha }$. It follows from the discussion above that $W_{ϵ}\subseteq {\cup }_{m=0}^n\alpha (V_{\alpha }\cap S^m).$

Let ${\mathit{v}}_0=(v_0^1,\dots ,v_0^d)=T\alpha \left({\mathrm{v}}_0\right)\in {\mathbb{R}}^d\cong T_{\alpha \left(x_0\right)}{\mathbb{R}}^d$. The assumptions about the chart $\alpha :V_{\alpha }\to W_{\alpha },$ made above, imply that $v_0^{n+1}=\dots =v_0^d=0$. By construction, for every $\mathit{x}=(x^1,\dots ,x^d)\in W{ }_{ϵ},$ the coordinates $x^{n+1},\dots ,x^d$ do not vanish, and some of the coordinates $x^1,\dots ,x^n$ may also be non-zero. Therefore, for every $\mathit{x}=(x^1,\dots ,x^d)\in W{ }_{ϵ}$, a vector $\mathit{v}=(v^1,\dots ,v^d)\in T_{\mathit{x}}{\mathbb{R}}^d\cong {\mathbb{R}}^d$ such that $v^1=\dots =v^n=0$ is tangent to $\alpha (V_{\alpha }\cap S^m)$ for every $m\le n$. On the other hand, for every $m\ge n$, $W_{ϵ}\cap \alpha (V_{\alpha }\cap S^m)=\varnothing$.

Choose a function $f\in C^{\infty }\left({\mathbb{R}}^d\right)$ such that $f\left({\mathit{x}}_0\right)=1$ and $f\left(\mathit{x}\right)=0$ for every $\mathit{x}\notin W{ }_{ϵ}$, and consider a vector field Y on ${\mathbb{R}}^d$ given by

$$Y\left(\mathit{x}\right)=f\left(\mathit{x}\right)\frac{\partial }{{\partial x}^{n+1}}+f\left(\mathit{x}\right)\frac{\partial }{{\partial x}^{n+2}}+\cdots +f\left(\mathit{x}\right)\frac{\partial }{{\partial x}^d}$$

for every $\mathit{x}\in {\mathbb{R}}^d$. Since $f\in C^{\infty }\left({\mathbb{R}}^d\right),$ it follows that integral curves of Y have open domains. The assumption that $f\left(\mathit{x}\right)=0$ for every $\mathit{x}\notin W{ }_{ϵ}$ implies that the integral curves of Y originating in $W_{\alpha }\supseteq {\overline{W}}_{ϵ}$ are contained in $W_{\alpha }$. Therefore, the restriction $Y_{\mid W_{\alpha }}$ of Y to $W_{\alpha }$ is a vector field on $W_{\alpha }$. The push-forward ${\alpha }_{*}^{-1}{Y}_{\mid W_{\alpha }}$ by the diffeomorphism ${\alpha }^{-1}:W_{\alpha }\to V_{\alpha }$ is a vector field on $V_{\alpha },$ which can be extended to a vector field $X\in \mathfrak{X}\left(S\right)$ vanishing outside ${\alpha }^{-1}\left(W_{ϵ}\right)\subseteq V_{\alpha }$. Since $f\left(\alpha \left(x_0\right)\right)=1$, it follows that $X\left(x_0\right)={\mathrm{v}}_0$, which completes the proof. □

Corollary 1.

It follows from the above results that the connected components of strata of the depth function stratification of the manifold with corners S are orbits of the Lie algebra $\mathfrak{X}\left(S\right)$ of all vector fields on S. Hence, the depth function stratification of S is given by the partition $\mathfrak{M}\left(S\right)$ of S by orbits of $\mathfrak{X}\left(S\right)$.

8. Concluding Remarks and Future Research

The partition $\mathfrak{M}\left(S\right)$ of a subcartesian space S by smooth manifolds provides a measure for the applicability of differential geometric methods to the study of the geometry of S. If all manifolds in $\mathfrak{M}\left(S\right)$ are single points, then we cannot expect differential geometry to be an effective tool in the study of S. On the other extreme, if $\mathfrak{M}\left(S\right)$ contains only one manifold M, then the subcartesian space S coincides with the manifold $M,$ and it is a natural domain for differential geometric techniques.

Understanding the structure of the partition $\mathfrak{M}\left(S\right)$ of a subcartesian space S is an important part of investigations of a geometric structure of S. The aim of the subsequent paper [21] is to use the knowledge of applying the techniques discussed here to investigate the structure of the cotangent bundle of a subcartesian space. In particular, we expect to decode the subcartesian analog of the symplectic structure of the cotangent bundle of a manifold. Once we have an understanding of symplectic subcartesian spaces, we shall be able to be more effective in the study of the reduction in symmetries of classical and quantum mechanics, which is the problem that has led both authors to the theory of differential spaces [14,22].