Vector Fields and Flows on Subcartesian Spaces
This paper is part of a series of papers on differential geometry of ∞-ringed spaces. In this paper, we study vector fields and their flows on a class of singular spaces. Our class includes arbitrary subspaces of manifolds, as well as symplectic and contact quotients by actions of compact Lie groups...
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| description | This paper is part of a series of papers on differential geometry of ∞-ringed spaces. In this paper, we study vector fields and their flows on a class of singular spaces. Our class includes arbitrary subspaces of manifolds, as well as symplectic and contact quotients by actions of compact Lie groups. We show that derivations of the ∞-ring of global smooth functions integrate to smooth flows.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 19 (2023), 093, 17 pages
Vector Fields and Flows on Subcartesian Spaces
Yael KARSHON ab and Eugene LERMAN c
a) School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv, Israel
E-mail: yaelkarshon@tauex.tau.ac.il
b) Department of Mathematics, University of Toronto, Toronto, Ontario, Canada
c) Department of Mathematics, University of Illinois at Urbana-Champaign,
Urbana, Illinois, USA
E-mail: lerman@illinois.edu
Received July 21, 2023, in final form November 08, 2023; Published online November 16, 2023
https://doi.org/10.3842/SIGMA.2023.093
Abstract. This paper is part of a series of papers on differential geometry of C∞-ringed
spaces. In this paper, we study vector fields and their flows on a class of singular spaces. Our
class includes arbitrary subspaces of manifolds, as well as symplectic and contact quotients
by actions of compact Lie groups. We show that derivations of the C∞-ring of global smooth
functions integrate to smooth flows.
Key words: differential space; C∞-ring; subcartesian; flow
2020 Mathematics Subject Classification: 58A40; 46E25; 14A99
1 Introduction
This paper is one in a series of papers on differential geometry of C∞-ringed spaces. Two other
papers in the series are [9] and [10].
In this paper, we study vector fields and flows on subcartesian spaces.1
Singular spaces, that is, spaces that are not manifolds, arise naturally in differential geometry
and in its applications to physics and engineering. There are many approaches to differential
geometry on singular spaces, and there is a vast literature which we will not attempt to survey.
In this paper we use differential spaces in the sense of Sikorski [15] as our model of singular
spaces.
Śniatycki’s book [18] contains a number of geometric tools that apply to differential spaces.
Śniatycki is particularly interested in stratified spaces that arise through symplectic reduction
(see [16]); he provides a new perspective by viewing these spaces as differential spaces. In this
paper, we respond to, and elaborate on, Śniatycki’s treatment of vector fields on differential
spaces. Specifically, for a derivation of (the ring of global smooth functions on) a subcartesian
space M , Śniatycki proves the existence and uniqueness of smooth maximal integral curves (see
[18, Theorem 3.2.1] and [17, Theorem 1]), but he does not discuss the flow as a map from
a subset of M × R to M .
The main result of this paper is Theorem 3.14, which roughly says the following:
Theorem. Let M be a differential space embeddable in some Euclidean space RN and v a vector
field on M . Assemble the maximal integral curves of v into a flow Φ: W → M , where W is
a subset M × R. Then the flow Φ: W → M is smooth.
1For us a vector field is a derivation of the C∞-ring of global smooth functions. This is different from Śniatycki’s
definition; see Remark 3.11. To reduce ambiguity, in our formal statements we say “derivation” rather than “vector
field”.
mailto:yaelkarshon@tauex.tau.ac.il
mailto:lerman@illinois.edu
https://doi.org/10.3842/SIGMA.2023.093
2 Y. Karshon and E. Lerman
Thanks to an analogue of the Whitney embedding theorem for differential spaces, embed-
dability in a Euclidean space is a fairly mild assumption on a differential space that is locally em-
beddable in a Euclidean space, i.e., the space that is subcartesian (see Definition 2.37). See [1, 11]
or [6] for various versions of the Whitney embedding theorem for subcartesian spaces. On the
other hand, if a differential space is not subcartesian, then the flow of a vector field may not
exist at all, see [7, Example 2, Section 32.12].
Theorem 3.14 relies on the existence and uniqueness of maximal integral curves. A few
years ago Śniatycki gave a proof of existence and uniqueness of integral curves of vector fields
on arbitrary subcartesian differential spaces (see [18, Theorem 3.2.1]). In a later paper [3],
Cushman and Śniatycki have a similar theorem, Theorem 5.3, and they say that “Theorem 5.3
replaces [5, Theorem 3.2.1], which is incorrect” (their [5] is our [18]). However, it seems to us
that there is nothing wrong with Śniatycki’s Theorem 3.2.1, certainly not with its statement.
To make sure, we provide a self-contained proof of existence and uniqueness of integral curves,
under the mild assumptions that imply embeddability, see Corollary 3.22.
In a later paper [5], we remove the mild assumptions that imply embeddability. These
assumptions are indeed mild: “reasonable” subcartesian spaces are embeddable. And removing
these assumptions has a price; the proof becomes more involved: for embeddable spaces, we can
rely on the integration of vector fields on open subsets of Euclidean spaces; for not-necessarily-
embeddable spaces, we need to imitate the proof of integration of vector fields on manifolds.
Organization of the paper
In Section 2, we recall the definition and some properties of differential spaces. This material
is standard. One novelty is that we explicitly mention C∞-rings. In Section 3, we prove the
existence and uniqueness of integral curves of derivations on embeddable subcartesian spaces
and use this result to prove the main theorem of the paper. In Appendix A, we provide a proof
of a special case of a theorem of Yamashita [19, Theorem 3.1]. Namely, we prove that any R-
algebra derivation of a point-determined C∞-ring is automatically a C∞-ring derivation. This
fact is used in our proof of the existence and uniqueness of integral curves of derivations.
Assumptions
Throughout the paper, “manifold” means “smooth (i.e., C∞) manifold”. All manifolds are
assumed to be second countable and Hausdorff.
2 Differential spaces
In this section, we recall the definition and some properties of differential spaces in the sense of
Sikorski. It will be convenient to recall the notion of a C∞-ring first. The definition below is not
standard, but it is easier to understand on the first pass. It is equivalent to Lawvere’s original
definition; see [4].
Definition 2.1. A C∞-ring is a set C , equipped with operations
gC : Cm → C
for all m ∈ Z≥0 and all g ∈ C∞(Rm), such that the following holds:
� For all n,m ∈ Z≥0, all f1, . . . , fm ∈ C∞(Rn) and g ∈ C∞(Rm),
(g ◦ (f1, . . . , fm))C (c1, . . . , cn) = gC ((f1)C (c1, . . . , cn), . . . , (fm)C (c1, . . . , cn))
for all (c1, . . . , cn) ∈ C n.
Vector Fields and Flows on Subcartesian Spaces 3
� For every m > 0 and for every coordinate function xj : Rm → R, 1 ≤ j ≤ m,
(xj)C (c1, . . . , cm) = cj .
If m = 0, then C 0 is a singleton {∗}. Similarly, C∞(
R0
)
≃ C∞(0) ≃ R. Thus 0-ary operations
on C are maps gC : {∗} → C , one for every g ∈ R. Since any map h : {∗} → C can be identified
with h(∗) ∈ C , we identify the 0-ary operation corresponding to g ∈ R with an element of C ,
which we denote by gC .
Example 2.2. Let M be a C∞-manifold and C∞(M) the set of smooth (real-valued) functions.
Then C∞(M), equipped with the usual composition operations
gC∞(M)(a1, . . . , am) := g ◦ (a1, . . . , am),
is a C∞ ring.
Example 2.3. Let M be a topological space and C0(M) the set of continuous real-valued
functions. Then C0(M), equipped with the usual composition operations
gC∞(M)(a1, . . . , am) := g ◦ (a1, . . . , am),
is also a C∞ ring.
Definition 2.4. A nonempty subset C of a C∞-ring A is a C∞-subring if C is closed under
the operations of A .
Example 2.5. If M is a manifold, then C∞(M) is a C∞-subring of C0(M).
We also need to recall the notion of an initial topology.
Definition 2.6. Let X be a set and F a set of maps from X to various topological spaces. The
smallest topology on X making all functions in F continuous is called initial.
In particular, a collection of real-valued functions F on a set X uniquely defines an initial
topology on X (we give the real line R the standard topology, of course).
Next we define differential spaces in the sense of Sikorski. The definition below agrees with
the one in [18]. Some papers define differential spaces as ringed spaces; see [14], for example.
Definition 2.7. A differential space (in the sense of Sikorski) is a pair (M,F ), where M is
a topological space and F is a (nonempty) set of real-valued functions on M , subject to the
following three conditions:
(1) The topology on M is the smallest topology making every function in F continuous, i.e.,
it is the initial topology defined by the set F .
(2) For any nonnegative integer m, any smooth function g ∈ C∞(Rm), and any m-tuple
f1, . . . , fm ∈ F , the composite g ◦ (f1, . . . , fm) is in F .
(3) Let g : M → R be a function. Suppose that for each point p of M there exist a neigh-
borhood U of p and a function a ∈ F such that g|U = a|U . Then the function g is
in F .
We refer to F as a differential structure on M .
Remark 2.8.
� We think of the set of functions F on a differential space (M,F ) as “smooth functions
by fiat”. (Also, see Remark 2.17.)
4 Y. Karshon and E. Lerman
� We may refer to a differential space (M,F ) simply as M .
� Condition (2) says that F is a C∞-ring with the operations gF : Fm → F given by
composition
gF (f1, . . . , fm) := g ◦ (f1, . . . , fm).
Note that since C∞(
R1
)
includes constant functions, (2) implies that all constant func-
tions are in F . Recall that 0-ary operations on a C∞-ring are indexed by constants
g ∈ C∞(
R0
)
≃ R. Given g ∈ C∞(
R0
)
we define the operation gF : F 0 = {∗} → F by
setting gF (∗) to be the constant function on M taking the value g everywhere. We know
that such a constant function has to be in F .
Remark 2.9. In the literature, the term “differential space” is used for a variety of mathematical
objects, some of which are related to Sikorski’s differential spaces, and some that are not related
at all.
Example 2.10. Let M be a manifold (second countable and Hausdorff). Then the pair
(M,C∞(M)), where C∞(M) is the set of C∞ functions, is a differential space in the sense
of Definition 2.7. The main point to check is that the topology on M coincides with the small-
est topology making all the functions in C∞(M) continuous. This follows from the existence
of bump functions on manifolds and from Lemma 2.15 below. Alternatively, it follows from
a theorem of Whitney by which any closed subset of a manifold M is the zero set of a smooth
function. See, for example, [8, Theorem 2.29].
Definition 2.11. Given a manifold M we refer to the C∞-ring C∞(M) of smooth functions
on M as the standard differential structure.
Example 2.12. Let M be a manifold. Then the set C0(M) of continuous function on M is
also a differential structure. Unless M is discrete, the C∞-ring C0(M) is bigger than C∞(M).
Definition 2.13. Let (M,T ) be a topological space, C ⊂ M a closed set and x ∈ M \ C
a point. A bump function (relative to C and x) is a continuous function ρ : M → [0, 1] so that
(supp ρ) ∩ C = ∅ and ρ is identically 1 on a neighborhood of x.
Definition 2.14. Let (M,T ) be a topological space and F ⊆ C0(M,R) a collection of contin-
uous real-valued functions on M . The topology T on M is F -regular iff for any closed subset C
of M and any point x ∈ M \ C there is a bump function ρ ∈ F with supp ρ ⊂ M \ C and ρ
identically 1 on a neighborhood of x.
Lemma 2.15. Let (M,T ) be a topological space and F ⊂ C0(M,R) a C∞-subring. Then T
is the smallest topology making all the functions in F continuous if and only if the topology T
is F -regular.
Proof. Let TF denote the smallest topology making all the functions in F continuous. The
set
S :=
{
f−1(I) | f ∈ F , I is an open interval
}
is a sub-basis for TF . Since all the functions in F are continuous with respect to T , TF ⊆ T .
Therefore, it is enough to argue that T ⊆ TF if and only if T is F -regular.
(⇒) Suppose T ⊆ TF . Let C ⊂ M be T -closed and x a point in M which is not in C.
Then M \ C is T -open. Since T ⊆ TF by assumption, M \ C is in TF . Then there exist
functions h1, . . . , hk ∈ F and open intervals I1, . . . , Ik such that x ∈ ∩k
i=1h
−1
i (Ii) ⊂ M \ C.
There is a C∞ function ρ : Rk → [0, 1] with supp ρ ⊂ I1 × · · · × Ik and the property that ρ = 1
Vector Fields and Flows on Subcartesian Spaces 5
on a neighborhood of (h1(x), . . . , hk(x)) in Rk. Then τ := ρ ◦ (h1, . . . , hk) is in F , since F is
a C∞-subring of C0(M). The function τ is a desired bump function.
(⇐) Suppose the topology T is F -regular. Let U ∈ T be an open set. Then C = M \ U
is closed. Since T is F -regular, for any point x ∈ U there is a bump function ρx ∈ F
with supp ρx ⊂ U and ρx is identically 1 in a neighborhood of x. Then ρ−1
x ((0,∞)) ⊂ U
and ρ−1
x ((0,∞)) ∈ TF . It follows that
U =
⋃
x∈U
ρ−1
x ((0,∞)) ∈ TF .
Since U is an arbitrary element of T , T ⊆ TF . ■
Definition 2.16. A smooth map from a differential space (M,FM ) to a differential space
(N,FN ) is a function φ : M → N such that for every f ∈ FN the composite f ◦ φ is in FM .
Remark 2.17. Given a differential space (M,F ), the set F coincides with the set of smooth
maps (M,F ) → (R, C∞(R)).
Remark 2.18. A map between two manifolds is smooth in the usual sense if and only if it is
a smooth map between the corresponding differential spaces (when both manifolds are given the
standard differential structures).
Remark 2.19. It is easy to see that the composite of two smooth maps between differential
spaces is again smooth. It is even easier to see that the identity map on a differential space is
smooth. Consequently, differential spaces form a category.
Definition 2.20. A smooth map between two differential spaces is a diffeomorphism if it is
invertible and the inverse is smooth.
Equivalently a smooth map is a diffeomorphism iff it is an isomorphism in the category of
differential spaces.
Remark 2.21. Every smooth map of differential spaces is continuous; this follows from Defini-
tion 2.7 (1).
Remark 2.22. Any differential structure F is an R-subalgebra of C0(M): for any f1, f2 ∈ F ,
λ, µ ∈ R
λf1 + µf2 = g ◦ (f1, f2), where g(x, y) := λx+ µy ∈ C∞(
R2
)
,
f1f2 = h ◦ (f1, f2), where h(x, y) := xy ∈ C∞(
R2
)
.
Remark 2.23. Any C∞-ring is an R-algebra (more precisely: has an underlying R-algebra struc-
ture). The binary operations + and · come from the functions h(x, y) = x+ y and g(x, y) = xy
respectively. The scalars come from the 0-ary operations.
We will not notationally distinguish between a C∞-ring and the corresponding (underlying)
R-algebra.
Definition 2.24. A differential structure F on a set M is generated by a subset A ⊆ F if F is
the smallest differential structure containing the set A. That is, if G is a differential structure
on M containing A, then F ⊆ G .
Lemma 2.25. Given a collection A of real-valued functions on a set M there is a differential
structure F on M generated by A. The initial topology for F is the initial topology for the
set A.
Proof. See [18, Theorem 2.1.7]. ■
6 Y. Karshon and E. Lerman
Notation 2.26. We write F = ⟨A⟩ if the differential structure F is generated by the set A.
Definition 2.27. Let (M,F ) be a differential space and N ⊆ M a subset. The subspace
differential structure FN on N , also known as the induced differential structure, is the differential
structure on N generated by the set A of restrictions to N of the functions in F :
A = {g : N → R | g = f |N for some f ∈ F}.
Definition 2.28. A smooth map f : (M,FM ) → (N,FN ) between two differential spaces is an
embedding if f is injective and the induced map f : (M,FM ) → (f(M), ⟨FN |f(M)⟩) from M to
its image (with the subspace differential structure) is a diffeomorphism.
Lemma 2.29. Let (M,F ) be a differential space and (N,FN ) a subset of M with the in-
duced/subspace differential structure. Then the smallest topology on N making all the functions
of FN continuous agrees with the subspace topology on N coming from the inclusion i : N ↪→ M .
Proof. The initial topology for the set F |N of generators of FN is the subspace topology.
Consequently, the initial topology for FN = ⟨F |N ⟩ is also the subspace topology (cf. Lem-
ma 2.25). ■
Remark 2.30. The subspace differential structure FN can be given a fairly explicit description:
FN =
{
f : N → R | there is a collection of sets {Ui}i∈I , open in M , with
⋃
i Ui ⊃ N
and a collection {gi}i∈I ⊆ F such that f |N∩Ui = gi|N∩Ui for all indices i
}
.
Remark 2.31. Let (M,F ) be a differential space and (N,FN ) a subset of M with the in-
duced/subspace differential structure. Then the inclusion map i : N ↪→ M is smooth since for
any f ∈ F , f ◦ i = f |N ∈ FN by definition of FN .
The subspace differential structure FN is the smallest differential structure on N making
the inclusion i : N → M smooth. This is because any differential structure G on N making
i : (N,G ) → (M,F ) smooth must contain the set F |N .
Lemma 2.32. Let (M,F ) be a differential space and (N,FN ) a subset of M with the in-
duced/subspace differential structure. For any differential space (Y,G ) and for any smooth
map φ : Y → M that factors through the inclusion i : N → M (i.e., φ(Y ) ⊂ N), the map
φ : (Y,G ) → (N,FN ) is smooth.
Proof. We need to show that φ∗h ≡ h ◦ φ ∈ G for any h ∈ FN . For any f ∈ F ,
G ∋ f ◦ φ = f ◦ i ◦ φ = φ∗(f |N ).
Consequently, φ∗(F |N ) ⊆ G . Since F |N generates FN and since G is a differential structure,
we must have φ∗FN ⊆ G as well. ■
Corollary 2.33. Let (M,F ) be a differential space and K ⊆ N ⊆ M subsets. Then
⟨F |K⟩ = ⟨⟨F |N ⟩|K⟩,
that is, the differential structure on K induced by the inclusion K ↪→ M agrees with the differ-
ential structure on K successively induced by the pair of the inclusions K ↪→ N ↪→ M .
Proof. Since for any f ∈ F , f |K = (f |N )|K ∈ ⟨F |N ⟩|K , we have F |K ⊆ ⟨⟨F |N ⟩|K⟩. Therefore,
⟨F |K⟩ ⊆ ⟨⟨F |N ⟩|K⟩.
On the other hand, the inclusion K ↪→ M factors through the inclusion K ↪→ N . Hence
by Lemma 2.32, the map j : (K, ⟨F |K⟩) ↪→ (N, ⟨F |N ⟩) is smooth. But the image of j lands
in K. Hence the identity map id: (K, ⟨F |K⟩) → (K, ⟨⟨F |N ⟩|K⟩) is smooth. Consequently,
⟨⟨F |N ⟩|K⟩ = id∗⟨⟨F |N ⟩|K⟩ ⊆ ⟨F |K⟩. ■
Vector Fields and Flows on Subcartesian Spaces 7
In the case where the differential space (M,F ) is a manifold and N is a subset of M the
subspace differential structure FN has a simple description.
Lemma 2.34. Let M be a manifold and N a subset of M , Then f : N → R is in C∞(M)N :=
⟨C∞(M)|N ⟩ (the subspace differential structure on N) if and only if there is an open neighbour-
hood U of N in M and a smooth function g : U → R such that f = g|N . Moreover, if N is
closed in M , we may take U = M .
Remark 2.35. Let M be a manifold and U ⊂ M an open subset. Then C∞(U) = ⟨C∞(M)|U ⟩.
This follows from the existence of bump functions.
Proof of Lemma 2.34. Let U ⊂ M be an open set with N ⊂ U , and let g ∈ C∞(U).
By Remark 2.35 and Corollary 2.33, for any open set U ⊂ M withN ⊂ U and any g ∈ C∞(U),
the restriction g|N is in C∞(M)N .
Conversely, suppose f ∈ C∞(M)N . Then there is an collection of open sets {Ui}i∈I with
N ⊂
⋃
i Ui and {gi}i∈I ⊂ C∞(M) so that f |Ui∩N = gi|Ui∩N for all i. Let U =
⋃
i Ui. There is
a partition of unity {ρi}i∈I on U subordinate to the cover {U}i∈I . Consider g :=
∑
ρigi∈C∞(U).
Then g|N = f .
If N is closed, then {Ui}i∈I ∪ {M \ N} is an open cover of M . Choose a partition of
unity {ρi}i∈I ∪ {ρ0} subordinate to this cover of M (with supp ρ0 ⊂ M \ N), and again set
g :=
∑
i∈I ρigi. Then g is a smooth function on all of M , and g|N = f . ■
Remark 2.36. Lemma 2.34 holds in greater generality. The proof does not really used the fact
that M is a manifold, it only needs the existence of the partition of unity. These do exist for
second countable Hausdorff locally compact differential spaces; see [18].
Definition 2.37. A differential space (M,F ) is subcartesian iff it is locally isomorphic to
a subset of a Euclidean (a.k.a. Cartesian) space: for every point p ∈ M , there is an open
neighborhood U of p in M and an embedding φ : (U,FU ) → (Rn, C∞(Rn)) (n depends on the
point p).
Products
The domain of a flow of a vector field on a manifold M is a subset of the product M × R.
Therefore, in order to define and understand flows of derivations on differential spaces we need
to understand finite products in the category of differential spaces.
Given two differential spaces (M1,F1) and (M2,F2) there are many differential structures
on their product M1 ×M2 so that the projections πi : M1 ×M2 → Mi, i = 1, 2 are smooth. The
smallest such structure is the one generated by the set π∗
1F1 ∪ π∗
2F2. We denote this structure
by Fprod. That is,
Fprod := ⟨π∗
1F1 ∪ π∗
2F2⟩.
Since the initial topology for π∗
1F1∪π∗
2F2 is the product topology, the initial topology for Fprod
is also the product topology (cf. Remark 2.25).
We next check that (M1 ×M2,Fprod) together with the projections π1, π2 has the universal
properties of the product in the category of differential spaces. Note that the projections π1, π2
are smooth.
Lemma 2.38. Let (M1,F1), (M2,F2) be two differential spaces, (Y,G ) another differential
space, φi : Y → Mi, i = 1, 2 two smooth maps. Then there exists a unique smooth map
φ : (Y,G ) → (M1 ×M2,Fprod) with πi ◦ φ = φi, i = 1, 2.
8 Y. Karshon and E. Lerman
Proof. Clearly there is a unique map of sets φ : Y → M1 × M2 with πi ◦ φ = φi, i = 1, 2.
Moreover since φi (i = 1, 2) are smooth
G ⊇ φ∗
iFi = φ∗(π∗Fi).
Therefore, φ∗(π∗
1F1 ∪π∗
2F2) ⊆ G . Since Fprod = ⟨π∗
1F1 ∪π∗
2F2⟩, φ∗Fprod ⊆ G as well. Thus φ
is smooth. ■
Remark 2.39. LetM1, M2 be two manifolds with the usual differential structures (i.e., C∞(M1)
and C∞(M2)). Then C∞(M1 ×M2) is the product differential structure on M1 ×M2.
We end the section by proving that taking products commutes with taking subspaces.
Lemma 2.40. Let (M1,F1), (M2,F2) be two differential spaces, N1 ⊆ M1, N2 ⊆ M2 two
subspaces, G1, G2 the subspace differential structures on N1, N2 respectively. Then the product
differential structure Gprod on N1 ×N2 is the subspace differential structure (Fprod)N1×N2.
Proof. It is enough to check that (N1 ×N2, (Fprod)N1×N2) together with the projections
pri : (N1 ×N2, (Fprod)N1×N2) → (Ni,Gi), i = 1, 2,
has the universal properties of the product (in the category of differential spaces).
We first argue that the projections pr1, pr2 are smooth. The projections
πi : (M1 ×M2,Fprod) → (Mi,Fi)
are smooth by definition of the product differential structure Fprod. Hence their restrictions
πi|N1×N2 : N1 ×N2 → Mi are smooth as well. Since πi(N1 ×N2) ⊆ Ni, the maps
pri = πi|N1×N2 : (N1 ×N2, (Fprod)N1×N2) → (Ni,Gi)
are also smooth (by Lemma 2.32).
Now let (Y,A ) be a differential space and φi : Y → Ni, i = 1, 2 be a pair of smooth maps.
Since the inclusions ȷi : Ni → Mi are smooth, the composites ȷi ◦ φi : Y → Mi, i = 1, 2, are
smooth. By the universal property of the product, there is a unique smooth map φ : (Y,A ) →
(M1×M2,Fprod) so that πi◦φ = ȷi◦φi. Consequently, (πi◦φ)(Y ) ⊆ Ni. Hence φ(Y ) ⊆ N1×N2.
Since N1×N2 is a subspace of (M1×M2,Fprod) the map φ : (Y,A ) → (N1×N2, (Fprod)N1×N2)
is smooth (see Lemma 2.32). Therefore, (N1 ×N2, (Fprod)N1×N2) together with the projections
pr1, pr2 is the product of (N1,G1) and (N2,G2). We conclude that (Fprod)N1×N2 = Gprod. ■
3 Derivations and their flows
A vector field v on a manifold M can be defined as a derivation v : C∞(M) → C∞(M) of the
R-algebra of smooth functions on M : v is R-linear and for any two functions f, g ∈ C∞(M) the
product rule holds:
v(fg) = v(f)g + fv(g).
One then proves that the chain rule also holds: for any n ≥ 1, any g ∈ C∞(Rn) and any
f1, . . . , fn ∈ C∞(M)
v(g ◦ (f1, . . . , fn)) =
n∑
i=1
((∂ig) ◦ (f1, . . . , fn)) · v(fi). (3.1)
Note that (3.1), appropriately interpreted, makes sense for any C∞-ring, since any C∞-ring is
an R-algebra (cf. Remark 2.23) and therefore carries the addition and multiplication operations.
Namely, we have the following definition.
Vector Fields and Flows on Subcartesian Spaces 9
Definition 3.1. Let A be a C∞-ring. A C∞-ring derivation of A is a function v : A → A so
that for any n ≥ 1, any g ∈ C∞(Rn) and any f1, . . . , fn ∈ A
v(gA (f1, . . . , fn)) =
n∑
i=1
((∂ig)A (f1, . . . , fn)) · v(fi).
It turns out, thanks to a theorem of Yamashita [19, Theorem 3.1], that any R-algebra deriva-
tion of a jet-determined C∞-ring is automatically a C∞-ring derivation. We will not explain
what “jet-determined” means, since this will take us too far afield. Suffices to say that all
C∞-rings arising as differential structures are jet-determined C∞-rings. In fact, there is a class
of C∞-rings that includes differential structures (namely for point-determined C∞-rings, see
Definition A.2) for which Yamashita’s theorem has a short proof. We present the proof in
Appendix A. From now on when talking about derivations of differential structures we will not
distinguish between R-algebra derivations and C∞-ring derivations since they are one and the
same.
Remark 3.2. Given a differential space (M,F ) we view a derivation v : F → F as the correct
analogue of a vector field on (M,F ). Thus “vector fields” in the title of the paper are derivations
of differential structures. See also Remark 3.12.
We now define integral curves of a derivation.
Definition 3.3. An interval is a connected subset of the real line R.
Remark 3.4.
� By Definition 3.3 a single point is an interval.
� The induced differential structure on an interval I ⊂ R is the set of smooth functions
C∞(I) on I (note that C∞(I) makes sense in all cases: I is open, closed, half-closed or
a single point).
� Unless the interval I is a singleton, there is a canonical derivation d
dx : C
∞(I) → C∞(I)
since we can differentiate smooth functions on an interval.
Definition 3.5. Let v : F → F be a derivation on a differential space (M,F ). An inte-
gral curve γ of v is either a map γ : {∗} → M from a 1-point interval or a smooth map
γ : (J,C∞(J)) → (M,F ) from an interval J ⊂ R so that
d
dx
(f ◦ γ) = v(f) ◦ γ
for all functions f ∈ F .
The curve γ starts at a point p ∈ M if 0 ∈ J and γ(0) = p.
Remark 3.6. We tacitly assume that all integral curves contain zero in their domain of defini-
tion. Thus any integral curve γ of a derivation starts at γ(0).
Definition 3.7. An integral curve γ : I → M of a derivation v on a differential space (M,F )
is maximal if for any other integral curve τ : K → M of v with τ(0) = γ(0) we have K ⊆ I
and γ|K = τ .
Remark 3.8. Note that maximal integral curves are necessarily unique.
The following examples are meant to illustrate two points:
(1) curves that only exist for time 0 should be allowed as integral curves of derivations and
(2) we should not require that the domain of an integral curve is an open interval.
10 Y. Karshon and E. Lerman
Example 3.9. Consider the derivation v = d
dx on the interval [0, 1]. Then γ : [−1/2, 1/2] →
[0, 1], γ(t) = t+ 1/2 is an integral curve of v. The curve γ is a maximal integral curve of v and
its domain is a closed interval.
Example 3.10. Let M be the standard closed disk in R2 : M =
{
(x, y) | x2+y2 ≤ 1
}
. Then M
is a manifold with boundary and a differential subspace of R2 (the two spaces of smooth functions
are the same!). Consider the vector field v = ∂
∂x on M . The curve γ : {0} → M , γ(0) = (0, 1)
is an integral curve of v; it only exists for zero time. The derivation v does have a flow in the
sense of Definition 3.13 below. The flow is
Φ: U ≡
{
((x, y), t) ∈ R2 × R | x2 + y2 ≤ 1, (x+ t)2 + y2 ≤ 1
}
→ M,
Φ((x, y), t) = (x+ t, y).
Note that while M is a manifold with boundary, the flow domain U is not a manifold with
boundary nor a manifold with corners. The domain U is a differential space, and the flow Φ is
smooth since it is the restriction to U of the smooth map Ψ: R3 → R2, Ψ((x, y), t) = (x+ t, y).
Remark 3.11. Śniatycki defines a vector field on a differential space (M,F ) to be a derivation
of F that integrates to local diffeomorphisms of M ; see [18, Definition 3.2.2]. In particular, the
domains of its maximal integral curves include neighbourhoods of 0. By this definition, a vector
field on a manifold with boundary must be tangent to the boundary.
Remark 3.12. We were tempted to call all derivations on differential spaces “vector fields”. In
the end we decided against it to avoid the clash with Śniatycki’s terminology.
Definition 3.13. Let v : F → F be a derivation on a differential space (M,F ). A flow of v is
a smooth map Φ: W → M from a subspace W of M × R with M × {0} ⊂ W such that for all
x ∈ M
(1) Φ(x, 0) = x;
(2) the set Ix := {t ∈ R | (x, t) ∈ W} is connected;
(3) the map Φ(x, ·) : Ix → M is a maximal integral curve for v (see Definition 3.7).
We are now in position to state the main result of the paper.
Theorem 3.14. Let (M,F ) be a differential space which is diffeomorphic to a subset of some Rn,
and v : F → F a derivation. Then v has a unique flow (see Definition 3.13).
Remark 3.15. The conditions of Theorem 3.14 are not as restrictive as they may seem at
the first glance since there a version of Whitney embedding theorem for subcartesian spaces
[1, 6, 11].2
2To precisely state the embedding theorem of Breuer, Marshall, Kowalczyk and Motreanu we need to recall
the definition of the structural dimension of a subcartesian space. It proceeds as follows: given a subcartesian
space M its structural dimension at a point x ∈ M is the smallest integer nx so that a neighborhood of x can be
embedded in Rnx . The structural dimension of a subcartesian space M is the supremum of the set of structural
dimensions of points of M . The embedding theorem (see [1, Theorem 2.2]) then says:
Theorem 3.16. Any second countable subcartesian space M of finite structural dimension can be embedded in
some Euclidean space.
Remark 3.17. If M is a subset of Rn (with the subset differential structure), then M is subcartesian and second
countable, and its structural dimension is bounded above by n. So the conditions of Theorem 3.16 are necessary.
A subset M of Rn need not be locally compact. Note that the conditions of Theorem 3.16 do not require local
compactness.
Remark 3.18. The disjoint union
⊔
n≥0 R
n is an example of a subcartesian space that is not embeddable in RN
for any N : its structural dimension is infinite.
Vector Fields and Flows on Subcartesian Spaces 11
On the other hand if the differential space is not subcartesian, then a derivation may have
infinitely many integral curves starting at a given point: see [7, Example 2, Section 32.12]. Such
a derivations does not have a flow. We are grateful to Wilmer Smilde for bringing this example
to our attention.
We start the proof of Theorem 3.14 by proving existence and uniqueness of maximal integral
curves. This, in turn, needs a lemma.
Lemma 3.19. Let A be a C∞-ring, w : A → A a derivation and a, b ∈ A two elements with
ab = 1 (i.e., a is invertible and b is the inverse of a). Then
w(b) = −b2w(a).
Proof. Since w is a derivation, w(1) = 0. Hence 0 = w(ab) = w(a)b + aw(b) and the result
follows. ■
Lemma 3.20. Let M ⊂ Rn be a subset with the induced differential structure F , W ⊂ Rn an
open neighborhood of M and v : F → F a derivation. Then for any function h ∈ C∞(W )
v(h|M ) =
n∑
i=1
(∂ih)|M · v(xi|M ), (3.2)
where x1, . . . , xn : Rn → R are the standard coordinate functions.
Proof. If h = k|W for some function k ∈ C∞(Rn), then
h|M = k|M = (k ◦ (x1, . . . , xn))|M = kF (x1|M , . . . xn|M ).
Hence, since v is a C∞-ring derivation,
v(h|M ) = v(kF (x1|M , . . . , xn|M )) =
n∑
i=1
(∂ik)|M · v(xi|M ) =
n∑
i=1
(∂ih)|M · v(xi|M )
and (3.2) holds for such a function h.
Otherwise by the localization theorem [12]3 there exist functions k, ℓ ∈ C∞(Rn) with ℓ|W
invertible in C∞(W ) so that h = k|W
ℓ|W . Then h|M = k|M (ℓ|M )−1 and therefore,
v(h|M ) = v
(
k|M (ℓ|M )−1
)
= v(k|M ) · (ℓ|M )−1 − k|M (ℓ|M )−2v(ℓ|M )
(by Lemma 3.19)
=
∑
i
(
(∂ik)|M (ℓ|M )−1 − k|M (ℓ|M )−2(∂iℓ)|M
)
· v(xi|M )
(since (3.2) holds for k|M and ℓ|M )
=
∑
i
∂i
(
k|W
ℓ|W
)∣∣∣∣
M
· v(xi|M ) =
n∑
i=1
(∂ih)|M · v(xi|M ). ■
Lemma 3.21. Let M ⊂ Rn be a subset, F the induced differential structure on M and
v : F → F a derivation. For any point p ∈ M , there exists a unique maximal integral curve
γ : I → M of v with γ(0) = p.
3For an exposition of this proof in English, see [13].
12 Y. Karshon and E. Lerman
Proof. By definition of the induced differential structure F on M , the restrictions xi|M ,
1 ≤ i ≤ n, are in F . Here as before x1, . . . , xn : Rn → R are the standard coordinate func-
tions. Then the functions v(xi|M ) are also in F , so there are open neighborhoods Ui of M in Rn
and bi ∈ C∞(Ui) with bi|M = v(xi|M ) (cf. Lemma 2.34). Let U =
⋂
1≤i≤n Ui, V :=
∑n
i bi
∂
∂xi
.
Then V is a vector field on U . Let γ̃ : J → Rn be the unique maximal integral curve of the
vector field V with γ(0) = p. Let I denote the connected component of the set (γ̃)−1(M) that
contains 0. We now argue that γ := γ̃|I is the desired maximal integral curve of the derivation v.
Note that since the image of γ lands in M , the map γ is smooth as a map from (I, C∞(I)) into
the differential subspace (M,F ) (see Lemma 2.32).
If I is the singleton {0}, there is nothing to prove. So suppose I ̸= {0}. Given f ∈ F there
is an open neighborhood W of M in Rn and a smooth function h ∈ C∞(W ) with f = h|M (see
Lemma 2.34). By replacing W with W ∩U if necessary, we may assume that W ⊂ U . Note that
for any t ∈ I, γ(t) = γ̃(t). We now compute
d
dt
(f ◦ γ)(t) = d
dt
(h ◦ γ̃)(t) = V (h)(γ(t))
(since γ̃ is an integral curve of V )
=
∑
i
(∂ih)(γ̃(t)) · bi(γ̃(t))
(by definition of V )
=
∑
i
(∂ih)(γ(t)) · v(xi|M )(γ(t))
(by definition of bi’s)
= v(h|M )(γ(t))
(by (3.2))
= v(f)(γ(t)).
Since f ∈ F is arbitrary, the curve γ is an integral curve of the derivation v.
We now argue that γ is a maximal integral curve of v. Let σ : K → M be another integral
curve of v with σ(0) = p. We first check that σ is an integral curve of the vector field V on W .
Note that since the inclusion M ↪→ W is smooth, σ : K → W is smooth. Consider h ∈ C∞(W ).
Then for any t ∈ K
d
dt
(h ◦ σ)(t) = d
dt
((h|M ) ◦ σ)(t) = v(h|M )(σ(t))
(since σ is an integral curve of v)
=
∑
i
(∂ih)|M (σ(t)) · v(xi|M )(σ(t))
(by (3.2))
=
∑
i
(∂ih)(σ(t)) · bi(σ(t))
Vector Fields and Flows on Subcartesian Spaces 13
(by definition of bi’s)
= V (h)(σ(t)).
Hence σ is an integral curve of the vector field V as claimed.
Since γ̃ is the maximal integral curve of V , σ = γ̃|K and K ⊂ γ−1(M). Since 0 ∈ K, K is
connected and since I is the connected component of 0 in γ−1(M), K ⊂ I. It follows that
σ = (γ̃|I)|K = γ|K and therefore γ = γ̃|I is the maximal integral curve of the derivation v. ■
We record two corollaries.
Corollary 3.22. Let (M,F ) be a second countable subcartesian space of bounded dimension
(so that the assumptions of the Whitney embedding theorem for subcartesian spaces apply, see
Remark 3.15 and the footnote). Then for any derivation v : F → F and for any point p ∈ M
there is a unique maximal integral curve γp of v with γp(0) = p.
The second corollary is really the corollary of the proof of Lemma 3.21.
Corollary 3.23. Let M ⊂ Rn be a subset, F induced differential structure on M and v : F → F
a derivation. There exists an open neighborhood U of M in Rn and a vector field V on U so that
for any p ∈ M the maximal integral curve γp : Ip → M of v with γp(0) = p is of the form γ̃p|Ip
for the maximal integral curve γ̃p : Jp → U of V with γ̃(0) = p.
Proof of Theorem 3.14. It is no loss of generality to assume that M ⊂ Rn and that the
differential structure F on M is the subspace differential structure: F = ⟨C∞(Rn)|M ⟩. By
Corollary 3.23, there is an open neighborhood U of M in Rn and a vector field V on U so that
for any p ∈ M the maximal integral curve γp : Ip → M of v with γp(0) = p is of the form γ̃p|Ip
for the maximal integral curve γ̃p : Jp → U of V . Let
W =
⋃
p∈M
{p} × Ip ⊂ M × R ⊂ U × R.
Note that by definition M × {0} ⊂ W . Define the map Φ: W → M by Φ(p, t) = γp(t) for
all (p, t) ∈ W . Then Φ is a flow of v modulo the issue of smoothness which we now address.
The vector field V on U has the flow
Ψ: W̃ → U, Ψ(x, t) := γ̃x(t),
where, as above, γ̃x : Jx → U is the maximal integral curve of V with γ̃(0) = x and
W̃ :=
⋃
x∈U
{x} × Jx.
For any point p ∈ M ,
Ψ|{p}×Ix = Φ|{p}×Ix
(since γ̃p|Ip = γp). Therefore, Φ = Ψ|W , hence smooth with respect to the differential structure
⟨C∞(U × R)|W ⟩ on W induced by the inclusion W ↪→ U × R.
It remains to show that ⟨C∞(U×R)|W ⟩ = ⟨Fprod|W ⟩, where Fprod is the product differential
structure on M × R. Note that Fprod depends only on the differential structures on M and R.
By Lemma 2.40, Fprod = ⟨C∞(U × R)|M×R⟩. By Corollary 2.33,
⟨⟨C∞(U × R)|M×R⟩|W ⟩ = ⟨C∞(U × R)|W ⟩.
Therefore, ⟨Fprod|W ⟩ = ⟨C∞(U × R)|W ⟩. ■
14 Y. Karshon and E. Lerman
Example 3.24. Let (M,ω) be a symplectic manifold with a Hamiltonian action of a compact
Lie group G and let µ : M → g∗ denote the corresponding equivariant moment map. Assume
that the action of G on M has only finitely many orbit types (this is the case, for example,
when M is the cotangent bundle of a compact manifold or when M itself is compact). Recall
that the symplectic quotient at 0 ∈ g∗ is defined to be the subquotient
M0 := µ−1(0)/G.
Let π : µ−1(0) → M0 denote the quotient map. The symplectic quotient M0 can be given the
structure of a differential space. Namely let C∞(M)G denote the space of G-invariant functions.
It is easily seen to be a C∞-subring of C∞(M). We define
F :=
{
f : M0 → R | f ◦ π = f̃ |µ−1(0) for some f̃ ∈ C∞(M)G
}
.
This idea goes back to the work of Cushman [2]. It is not hard to check that F is a differential
structure on M0. For instance, this follows from the existence of the desired bump functions.
See also [18].
By [16, Example 6.6], the differential space (M0,F ) is embeddable. Consequently, any
derivation of F has a unique smooth flow.
A R-algebra and C∞-ring derivations of differential structures
The goal of this appendix is to prove that for any point-determined C∞-ring C (see Defini-
tion A.2) any R-algebra derivation v : C → C is automatically a C∞-ring derivation. We start
by defining R-points of C∞-rings.
Definition A.1. An R-point of a C∞-ring C is a nonzero homomorphism φ : C → R of C∞-
rings.
Definition A.2. A C∞-ring C is point-determined if R-points separate elements of the ring.
That is for any a ∈ C , a ̸= 0 there is an R-point φ : C → R with φ(a) ̸= 0.
Example A.3. Let (M,F ) be a differential space and x ∈ M a point. Then the evaluation
map
evx : F → R, evx(f) := f(x)
is an R-point of F . The C∞-ring F is point-determined since for any nonzero function f ∈ F
there is a point x ∈ M with 0 ̸= f(x) = evx(f).
We next recall Hadamard’s lemma.
Lemma A.4 (Hadamard’s lemma). For any smooth function f : Rn → R, there exist (non-
unique) smooth functions g1, . . . , gn ∈ C∞(Rn × Rn) such that
f(x)− f(y) =
n∑
i=1
(xi − yi)gi(x, y)
for any pair of points x, y ∈ Rn.
Moreover, for n-tuple of functions h1, . . . , hn ∈ C∞(Rn × Rn) with the property that
f(x)− f(y) =
n∑
i=1
(xi − yi)hi(x, y)
for all (x, y) ∈ Rn × Rn we have
hi(b, b) = (∂if)(b)
for all b ∈ Rn.
Vector Fields and Flows on Subcartesian Spaces 15
Proof.
f(x)− f(y) =
∫ 1
0
d
dt
f(tx+ (1− t)y)dt =
∫ 1
0
n∑
i=1
∂if(tx+ (1− t)y)(xi − yi)dt
=
n∑
i=1
(xi − yi)
∫ 1
0
∂if(tx+ (1− t)y)dt.
Define
gi(x, y) =
∫ 1
0
∂if(tx+ (1− t)y)dt.
This proves existence of the desired functions g1, . . . , gn. To prove the second part of the lemma,
note that
(∂if)(b) = lim
s→0
1
s
(f(b+ sei)− f(b)),
where ei is the ith standard basis vector. Therefore if f(x)− f(y) =
∑n
i=1(xi − yi)hi(x, y), then
(∂if)(b) = lim
s→0
1
s
n∑
j=1
((b+ sei)j − bj)hj(b+ sei, b) = lim
s→0
1
s
shi(b+ sei, b) = hi(b, b). ■
We are now in position to prove the main result of the appendix.
Theorem A.5. Let A be a point-determined C∞ ring and v : A → A an R-algebra derivation.
Then v is a C∞-derivation.
Proof. Recall that if A is a unital R-algebra and v : A → A is an R-algebra derivation, then
v(1A ) = 0A since v(1A ) = v
(
12A
)
= 1A v(1A ) + v(1A )1A = v(1A ) + v(1A ).
Let h ∈ C∞(
Rk
)
be a smooth function and a1, . . . , ak ∈ A . Let x : A → R be an R-
point. Then b = (b1, . . . , bk) := (x(a1), . . . , x(ak)) is a point in Rk. By Hadamard’s lemma (see
Lemma A.4), there are smooth functions g1, . . . , gk ∈ C∞(
R2k
)
such that
h(y) = h(b) +
k∑
j=1
(yj − bj)gj(y, b)
for all y ∈ Rk, and gj(b, b) = ∂jh(b). Let ĝj(y) := gj(y, b). Then, for any (a1, . . . , ak) ∈ A k,
hA (a1, . . . , ak) = h(b)A +
k∑
j=1
(aj − (bj)A ) · (ĝj)A (a1, . . . , ak).
Applying the algebraic derivation v to both sides and using the fact that v applied to a scalar
is zero, we get
v(hA (a1, . . . , ak)) =
k∑
j=1
v(aj)(ĝj)A (a1, . . . , ak) +
k∑
j=1
(aj − (bj)A )v((ĝj)A (a1, . . . , ak).
Now we apply the R-point x to both sides and use the fact that x(aj − (bj)A ) = x(aj)− bj = 0
for all j. We get
x(v(hA (a1, . . . , ak)) =
k∑
j=1
x(v(aj)) · x
(
(ĝj)A (a1, . . . , ak)
)
.
16 Y. Karshon and E. Lerman
Finally, note that for each j,
x
(
(ĝj)A (a1, . . . , ak)
)
= (ĝj)C∞(R)
(
x(a1), . . . , x(ak)
)
(since x is a homomorphism of C∞-rings)
= gj(b1, . . . , bk, b1, . . . , bk)
= (∂jh)(b1, . . . , bk) = (∂jh)(x(a1), . . . , x(ak))
= x
(
(∂jh)A (a1, . . . , ak)
)
(since x is a homomorphism of C∞-rings). Therefore
x
(
v(hA (a1, . . . , ak))
)
=
k∑
j=1
x
(
v(aj)
)
x
(
(∂jh)A (a1, . . . , ak)
)
= x
( k∑
j=1
(∂jh)A (a1, . . . , ak)v(aj)
)
.
Since A is point determined and since the R-point x is arbitrary,
v
(
hA (a1, . . . , ak)
)
=
k∑
j=1
(∂jh)A (a1, . . . , ak)v(aj),
i.e., v is a C∞-ring derivation. ■
Acknowledgements
We thank Jordan Watts and Rui Fernandes for their help. Y.K.’s research is partly funded by
the Natural Science and Engineering Research Council of Canada and by the United States –
Israel Binational Science Foundation. E.L.’s research is partially supported by the Air Force
Office of Scientific Research under award number FA9550-23-1-0337.
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1 Introduction
2 Differential spaces
3 Derivations and their flows
A R-algebra and C^infty-ring derivations of differential structures
References
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| id | nasplib_isofts_kiev_ua-123456789-212038 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-18T11:51:10Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Karshon, Yael Lerman, Eugene 2026-01-23T10:10:35Z 2023 Vector Fields and Flows on Subcartesian Spaces. Yael Karshon and Eugene Lerman. SIGMA 19 (2023), 093, 17 pages 1815-0659 2020 Mathematics Subject Classification: 58A40; 46E25; 14A99 arXiv:2307.10959 https://nasplib.isofts.kiev.ua/handle/123456789/212038 https://doi.org/10.3842/SIGMA.2023.093 This paper is part of a series of papers on differential geometry of ∞-ringed spaces. In this paper, we study vector fields and their flows on a class of singular spaces. Our class includes arbitrary subspaces of manifolds, as well as symplectic and contact quotients by actions of compact Lie groups. We show that derivations of the ∞-ring of global smooth functions integrate to smooth flows. We thank Jordan Watts and Rui Fernandes for their help. Y.K.’s research is partly funded by the Natural Science and Engineering Research Council of Canada and by the United States-Israel Binational Science Foundation. E.L.’s research is partially supported by the Air Force Office of Scientific Research under award number FA9550-23-1-0337. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Vector Fields and Flows on Subcartesian Spaces Article published earlier |
| spellingShingle | Vector Fields and Flows on Subcartesian Spaces Karshon, Yael Lerman, Eugene |
| title | Vector Fields and Flows on Subcartesian Spaces |
| title_full | Vector Fields and Flows on Subcartesian Spaces |
| title_fullStr | Vector Fields and Flows on Subcartesian Spaces |
| title_full_unstemmed | Vector Fields and Flows on Subcartesian Spaces |
| title_short | Vector Fields and Flows on Subcartesian Spaces |
| title_sort | vector fields and flows on subcartesian spaces |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212038 |
| work_keys_str_mv | AT karshonyael vectorfieldsandflowsonsubcartesianspaces AT lermaneugene vectorfieldsandflowsonsubcartesianspaces |