Quick question on Riemannian geometry

2010-12-25T17:41:14Z

Hey.

I got a quick question on Riemannian geometry. I'm not quite sure whether this is the right place to ask this question, since it might be a rather elementary one from a research point of view. Nontheless I think it might be too advanced for a typical homework question (which by the way it isn't)

So let $M^m$ be a manifold embedded into euclidean space $R^n$. Let L($\gamma$) be the the lenght of a smooth curve $\gamma$: [0,1] --> $R^n$ which is the number $\int_0^1 \ |d/dt \gamma(t)| \ \mathrm{d}t$ . Define the distance function d: M x M --> [0,$\infty$) by d(p,q):=inf L ($\gamma$). The infimum is taken over all smooth paths connecting p and q. This formula defines a metric on M.

I want to show that $\forall$ $p_0 \in M^m \exists U \subset M^m $ open neighborhood of $p_0$ such that (1-$\epsilon$) |p-q| < d(p,q) < (1+$\epsilon$) |p-q| . Note that |.| denotes the usual euclidean distance on $M^m$.

Now we fix an arbitrary $p_0 \in M^m$. Without loss of generality (Translations and rotations preserves the length of a curve) we assume that $p_0$ = 0 and $T_ {p_0} M$ = $R^m$ x {0} .

Now comes the step that I don't understand.

Now there's a smooth function f: $\Omega$ --> $R^{n-m}$ defined on an open neighborhood $\Omega \subset R^m$ of the origin such that { (x,y) $\in R^m x R^{n-m}$ | x $\in \Omega$ , y=f(x) } $\subset M^m$ , f(0)=0, df(0)=0

Now I don't really see where this is coming from. It seems to me that it might be some combination of a characterization of manifolds and tangent spaces we introduced, which is the following:

Let $M^m \subset R^n$, p $\in M^m$ "smooth manifold" , then $\exists U \subset R^n$ and f: U --> $R^{n-m}$ smooth such that p $\in$ U , df(q) surjective for all q $\in$ U $\cap$ M and U $\cap$ M = $f^{-1}$(0).

Can anyone help me?

Thanks in advance.

Quick question on Riemannian geometry - MathOverflow

Quick question on Riemannian geometry