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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?

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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.

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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! –  Don Toddo Feb 3 '12 at 15:21
    
..I meant to write "this NOW really is the..." :) –  Don Toddo Feb 3 '12 at 15:22
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