On the leaves of a prefoliation of a 𝕂-differential space

Andrzej Pia̧tkowski

March 7, 2001

Two years ago at the conference in Krynica, prof. W. Waliszewski has presented the definition of a 𝕂-differential space which is a generalization of the notions of a differentiable manifold and a differential space in the sense of Sikorski.

I would like to define the notion of a prefoliation of a 𝕂-differential space and to present theorems which describe some properties of leaves of the prefoliation.

First, I remind the definition of a 𝕂-differential space.

Let 𝕂 be an arbitrary field with a non-trivial norm (we can think here about ℝ or ℂ). Let M^{(0)} = {α : α : D_{α} → 𝕂} be a family of functions with an arbitrary family of sets {D_{α}} as domains. Define the set

.M^{(0)} = ⋃_{α∈M_{0}} D_{α}

which will be called the set of points of M^{(0)}. In the set of points of M^{(0)} define a topology top M^{(0)} as the weakest topology containing the family

{α^{−1}(B) : B is open in 𝕂 and α ∈ M^{(0)}}.

Next set

an M^{(0)} := {φ ∘ (α_{1}, ...α_{m}) : m ∈ ℕ and α_{1}, ..., α_{m} ∈ M^{(0)}
and φ is an analytical function defined on an open

set in 𝕂^{m} with values in 𝕂}.

If A ⊂ .M^{(0)} then

M^{(0)}|A := {α|A ∩ D_{α} : α ∈ M^{(0)}}

and

M_{A}^{(0)} := {β : ∀_{p∈D_{β}}∃_{U∈top M^{(0)}}∃_{α∈M^{(0)}}(p ∈ U ∩A ⊂ D_{β} ∧U ⊂ D_{α} ∧β|U ∩A = α|U ∩A)}.

It is easy to see that M^{(0)}|A ⊂ M_{A}^{(0)}.

Definition 1 The family M of functions with its values in 𝕂 is called a 𝕂-differential space, if the condition

an M = M = M_{.M}

is fulfilled.

Let M^{(0)} be an arbitrary family of functions with its values in 𝕂. One can prove the following

Proposition 2 The family M := (an M^{(0)})_{.M^{(0)}} is the smallest 𝕂-differential space with the set .M^{(0)} as the set of points, containing M^{(0)}.

Definition 3 The 𝕂-differential space M defined above is called the 𝕂-differential space generated by a family M^{(0)}.

Prof. W. Waliszewski has proved the following

Proposition 4 Let M be a 𝕂-differential space and A ⊂ .M. Then M_{A} ⊂ M if and only if A ∈ top M.

The above proposition gives

Corollary 5 Let M be a 𝕂-differential space and A ∈ top M. Then M|A = M_{A}.

Let M, N be 𝕂-differential spaces.

Definition 6 The mapping f : .M → .N is said to be smooth if for each β ∈ N we have β ∘ f ∈ M.

If the above condition holds then we write f : M → N.

It is obvious that if f : M → N then f : top M → top N i.e. f is a continuous mapping respective to the topologies top M and top N.

Let M be a 𝕂-differential space and p ∈ .M. Define M(p) := {α ∈ M : p ∈ D_{α}}.

Definition 7 Any 𝕂-linear mapping v : M(p) → 𝕂 such that

v(α · β) = v(α) · β(p) + α(p) · v(β)

for α, β ∈ M(p) is called a vector tangent to M at p. The family of all vectors tangent to M at p forms a vector space. This vector space is said to be tangent to M at p. It is denoted by T_{p} M.

It is easy to see that if v ∈ T_{p} M, U ∈ top M and α ∈ M then v(α) = v(α|U). Let M, N be 𝕂-differential spaces and f : M → N. For each p ∈ .M, the mapping

f determines a linear mapping (f_{∗})_{p} : T_{p} M → T_{f(p)} N called a tangent mapping. Namely, for v ∈ T_{p} M and β ∈ N(f(p)) we have

((f_{∗})_{p}v) (β) = v(β ∘ f).

Definition 8 The mapping f : M → N is called an immersion, if for each p ∈ .M the tangent mapping (f_{∗})_{p} is a monomorphism.

Now we define a prefoliation of a 𝕂-differential space. Let M = {α : α : D_{α} → 𝕂} be a 𝕂-differential space.

Definition 9 A pair (M, F) of 𝕂-differential spaces is called a prefoliation of M if

1) .F = .M,
2) top F is locally connected,
3) ∀_{p∈.M}∃_{U∈top F} (p ∈ U ∧ F_{U} = M_{U}).

Connected components of top F are called leaves of (M, F).

It is easy to see that the notion of a prefoliation is a generalization of the notion of the regular foliation, and of the Stefan foliation. Moreover, if (M, F) is a prefoliation of 𝕂-differential space M then (top M, top F) is a topological foliation in the sense of Ehresmann.

It is not difficult to prove the following

Theorem 10 If (M, F) is a prefoliation of a 𝕂-differential space M then for the mapping f = id_{.M} we have f : F → M.

Proof. If β ∈ M then for each p ∈ .M denote by U_{p} such a neighbourhood of p respective to top F for which

F_{U_{p}} = M_{U_{p}}.

Thus we get an open covering {U_{p}}_{p∈.M} of the set .M respective to top F with

β|U_{p} ∈ M|U_{p} ⊂ M_{U_{p}} = F_{U_{p}}.

Therefore β ∈ F, since F is a 𝕂-differential space.

From the theorem we get

Corollary 11 Let (M, F) be a prefoliation of a 𝕂-differential space M. Then top M ⊂ top F and M ⊂ F.

We also have

Theorem 12 If (M, F) is a prefoliation of a 𝕂-differential space M then the mapping f = id_{.M} is an immersion.

Proof. Suppose that (f_{∗})_{p}(v) = 0 for some v ∈ T_{p} F. Then for each β ∈ M(p) we have v(β ∘ f) = 0, i.e. for each G ∈ top F with p ∈ G we have

v((β ∘ f)|G) = 0  (1)

Let α ∈ F (p). There exists U ∈ top F such that p ∈ U and F_{U} = M_{U} by the definition of a prefoliation. Therefore, α|D_{α} ∩ U ∈ F |U = F_{U} = M_{U} by Corollary 5. Thus there exist V ∈ top M and γ ∈ M such that p ∈ V ∩ U ⊂ D_{α} and V ⊂ D_{γ} and γ|V ∩ U = α|V ∩ U. Obviously γ ∈ M(p) and because of Corollary 11, V ∩ U ∈ top F. Consequently,

v(α) = v(α|V ∩ U) = v(γ|V ∩ U) = v(γ ∘ f|V ∩ U) = 0

by (1). Thus v = 0.

Corollary 13 If L is a leaf of a prefoliation (M, F) and φ : L ∋ q ↦ q ∈ .M then φ : F_{L} → M and φ is an immersion.

Corollary 14 If L is a leaf of a prefoliation (M, F) then M_{L} ⊂ F |L.

We show that even if id : F → M is an immersion and (top M, top F) is a topological foliation then (M, F) has not to be a prefoliation.

Example 15 Let M be the ℝ-differential space generated by the family of all continuous functions defined on ℝ with values in ℝ and let F be the ℝ-differential space generated by the family of all C^{∞} functions defined on ℝ with values in ℝ. Then (top M, top F) is the trivial topological foliation in the sense of Ehresmann of ℝ. Remark that f = id_{ℝ} : ℝ → ℝ is a smooth mapping F → M since M ⊂ F. One can prove that T_{p} F is a vector space of dimension 0 for each p ∈ ℝ. Therefore, (f_{∗})_{p} is a monomorphism for each p ∈ ℝ.

It is obvious that for each U ∈ top F = top M we have M_{U} = M|U ≠ F |U = F_{U} which means that (M, F) is not a prefoliation of an ℝ-differential space M.

Using the definition of a prefoliation, one can prove

Lemma 16 If (M, F) is a prefoliation of a 𝕂-differential space, then for each β ∈ F and for each p ∈ D_{β} there exist U ∈ top F and α ∈ M such that p ∈ U ⊂ D_{β} and β|U = α|U.

Definition 17 The leaf L of a prefoliation (M, F) is said to be proper if (top F)|L = (top M)|L.

We have the following

Theorem 18 Let (M, F) be a prefoliation of a 𝕂-differential space M and L be a proper leaf of (M, F). Then

M_{L} = F |L.

Proof. The inclusion M_{L} ⊂ F |L holds for each L by Corollary 14.

Let γ ∈ F |L. By a local connectedness of top F we have L ∈ top F and consequently γ ∈ F |L ⊂ F by Proposition 4 and Corollary 5. By Lemma 16, for each p ∈ D_{γ} there exists V ∈ top F such that p ∈ V ⊂ D_{γ} ⊂ L and there exists α ∈ M such that γ|V = α|V. Since L is proper, there exists W ∈ top M such that V = W ∩ L. Let p ∈ D_{γ} and define U := W ∩ D_{α} ∈ top M and \overline{α} := α|U ∈ M. Then

1) p ∈ U ∩ L ⊂ D_{γ} since U ∩ L ⊂ W ∩ L = V ⊂ D_{γ};

2) U ⊂ D\overline{α} (in fact, the equality holds);

3) \overline{α}|U ∩ L = α|U ∩ L = α|W ∩ L ∩ D_{α} = γ|W ∩ L ∩ D_{α} = γ|U ∩ L since W ∩ L ∩ D_{α} ⊂ V.

By 1)-3) we have γ ∈ M_{L}.