2
$\begingroup$

In an article by Lubich, I came across a decomposition for points on the straight line between two points lying in an embedded submanifold $M$ of $R^{n}$.

To be precise, it is proposed that for $X$, $\tilde X \in M$ and $\tau$ small enough, any point $X + \tau(\tilde X -X)$ may be decomposed into $$ X + \tau(\tilde X -X) = Y(\tau) + Z(\tau)$$ with $Y \in M$ and $Z \bot T_XM$, the tangent space at $X$.

To me, this looks like an application of some implicit function theorem or something, but as I am no expert in manifold theory, I just cannot get my head around it. Is there anyone who can help?

$\endgroup$
2
$\begingroup$

In fact, any point $\tilde x$ near $x\in M$ can be decomposed like that. You can see this as follows: Locally $M\subset \mathbb{R}^n$ can be given as a graph: w.l.o.g. $x=0$ and there is a small ball $U$ around $x$ and a function $f\colon U\cap V\to W$ where $V\oplus W=\mathbb{R}^n$ is an decomposition into orthogonal linear subspaces such that $M\cap U=graph(f).$ Then $V=T_xM$ and with $\pi^V,\pi^W$ the orthogonal projections you obtain $$\tilde x=\pi^V(\tilde x)\oplus f(\pi^V(\tilde x))+0\oplus(\pi^W(\tilde x)-f(\pi^V(\tilde x)))$$ as desiered.

$\endgroup$
2
  • $\begingroup$ Thanks! I'll have to check on the existence of that $f$, but I guess this not really is the implicit function theorem I had in mind... A good weekend to you! $\endgroup$
    – Don Toddo
    Feb 3 '12 at 15:21
  • $\begingroup$ ..I meant to write "this NOW really is the..." :) $\endgroup$
    – Don Toddo
    Feb 3 '12 at 15:22

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.