Closed Maximal Integral Manifold is a regular submanifold.

Question

I am attempting the following problem from Boothby

Let N be a maximal integral manifold of a distribution Δ on M. Show that if N is closed (as a subset), then it is a regular submanifold of M"

My guess is that in order to show that it's a regular submanifold, I have to prove that the mapping $F: N \rightarrow M$ is an embedding. Since N is closed, all I have to show that it is bounded and thereby implying that the subspace is compact. I cannot seem to figure out how to prove this.

Also if this approach is wrong, please point me in the right direction. Thanks.

Answer 1

Here is my attempt:

Since N is an immersed submanifold, $i:N \hookrightarrow M$ is an injective immersion. Also, the topology of $i(N)$ is the subspace topology induced by M. Given that $i(N)$ is closed, i is a closed mapping and therefore a homeomorphism onto its image, and therefore is a $C^{\infty }$ embedding. Which proves that it is a regular submanifold of M.