Topology of the leaves on a manifold?

Question

Can anyone explain me the construction of the topology of the leaves of a foliation on a manifold and how the leaves are the connected components of this manifold with respect to that topology?

Futhermore, the manifold has already a topology, what is the relation between the original topology of the manifold and the topology of the leaves?

Thanks

Answer 1

What your alluding to is sometimes called the "leaf topology" on the manifold. To describe it, I'll use the approach of representing a foliation of $M$ by special "foliation coordinate charts".

If $M$ has dimension $k+l$, a foliation of $M$ of dimension $k$ can be described by charts which have the form $\phi_i : U_i \to V_i \times T_i$ where $U_i \subset M$ is open, $V_i \subset \mathbb{R}^k$ is open, and $T_i \subset \mathbb{R}^l$ is open. The overlap requirement is that if $p \in U_i \cap U_j$ then $U_i \cap U_j$ contains a neighborhood of $p$ of the form $\phi_i^{-1}(V'_i \times T'_i)$, where $V'_i \subset V_i$ is open and $T'_i \subset T_i$ is open, such that the overlap map $$\phi_j \circ \phi_i^{-1} \biggm| V'_i \times T'_i : V'_i \times T'_i \to V_j \times T_j $$ has the form $$F(v,t) = (f(v,t),g(t)) \in V_j \times T_j $$ Often there are also smoothness requirements on $f$, and sometimes on $g$ as well.

Subsets of $M$ of the form $\phi_i^{-1}(V'_i \times t)$, where $V'_i \subset V_i$ is open and $t \in T_i$, are sometimes called "plaques" of the foliation. The overlap requirement says, informally, that a small enough plaque in one foliation coordinate chart must be taken, via the overlap map, into a plaque in the other coordinate chart.

The set of all plaques forms the basis of a topology on $M$ called the "leaf topology". And in this topology, the connected components are indeed the leaves. Also, this "leaf topology" is the one with respect to which the each leaf is an "immersed sub manifold" in the answer of @JackLee.

Answer 2

This is somewhat complicated. The theory is explained in detail in my Introduction to Smooth Manifolds (2nd ed.), Chapter 19.

Basically, each leaf is an immersed submanifold. This is a subset, endowed with a topology (not necessarily the subspace topology) and a smooth structure under which the inclusion map is a smooth immersion. The construction of the topology and smooth structure is based on finding "local flat charts" for the foliation.

A partial answer to your second question is that the topology on the leaves is either the same as the subspace topology or strictly finer than the subspace topology. (This is a simple consequence of the fact that the inclusion of each leaf is continuous.) But there's quite a bit more to it than that.