0

I am trying to answer the following questions with a True (and give a proof) or a False (and give a counter example). I really have no idea how to approach this problems or how to start thinking about them. I would appreciate any hints, comments or suggestions on how to decide the answer.

a) Let $M$ and $N$ be embedded sub manifolds of $\mathbb{R}^3$. Then $M \cap N$ is an embedded sub manifold of $\mathbb{R}^3$ iff $M$ and $N$ intersect transversely.

b) Let $F: S^{35} \rightarrow \mathbb{R}^{36}$ be a smooth map. Then the image $F(S^{35})$ has measure zero in $\mathbb{R}^{36}$

f) If $F:M \rightarrow N$ is a submersion, then for all $p,q \in N$, them manifolds $F^{-1}(p)$ and $F^{-1}(q)$ are diffeomorphic.

Thanks for your help!

BOlivianoperuano84
  • 597

1 Answers1

1

Hints:

(a) Duplicate for the hard implication: see Why is a transversal intersection of submanifolds a manifold? The other implication is obviously...

(b) Think in the easier cases $F: S^1\rightarrow\mathbb{R}^2$ or $F: S^2\rightarrow\mathbb{R}^3$.

(c) What happens if $F^{-1}(p)\ne\emptyset$ and $F^{-1}(q) = \emptyset$?

Martín-Blas Pérez Pinilla
  • 42,722
  • 4
  • 51
  • 94