(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let's assume that M is a compact n-dimensional manifold,

then from Whitney's Immersion Theorem,

we know that there's an immersion, f: M -> R_2n, and

let's define f*: TM --> R_2n such that

f* sends (p, v) to df_x (v).

Since f is an immersion, it's clear that f* must be one-to-one by definition of immersion.

let "x" be a regular value of f*, then how would you show that

the inverse image of x (with respect to f, not f*) consists of finitely many points?

2. Relevant equations

3. The attempt at a solution

I reduced the problem to this. Once I show that there are only finitely many preimages (with respect to f, not f*) of

x in the compact set C = {(p, v) in TM : |v| <= 1}, I'm done.

I tried to prove it using proof by contradiction, so

I assumed that there are infinitely many points in that set, then we obviously,

there's a subsequence such that

pi --> p vi --> w for some (p, w) in C

by properties of a compact set.

Then once I show that df_p(w) = 0, I'm done, but

I'm struggling to show how that would work.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Immersion and Manifold Question

Can you offer guidance or do you also need help?

**Physics Forums | Science Articles, Homework Help, Discussion**