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Let $\mathcal M \subset \mathbb R^d$ be a smooth manifold, and for each $s \in \mathcal M$ let $T_s[\mathcal M]$ denote the tangent space of $\mathcal M$ at $s$. For each $s \in \mathcal M$ let $P_s$ denote the orthogonal projection of $\mathbb R^d$ into $T_s[\mathcal M]$. What can be said about the continuity properties of $P_s$ in $s$?

In particular I seek that is has a modulus of continuity.

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I would say, this map $s\mapsto P_s$ is smooth as $\mathcal M\to \Bbb R^{d\times d}$, assumed that the embedding $\mathcal M\hookrightarrow \Bbb R^d$ is smooth. –  Berci Nov 7 '12 at 16:58
    
What makes you say this? A sketch would suffice. –  dcs24 Nov 7 '12 at 18:18

1 Answer 1

up vote 2 down vote accepted

You want to show that the map $P:x\in M\to P_x\in L(\mathbb R^d,\mathbb R^d)$ is smooth.

Necessary and sufficient condition for the smoothness of $P:M\to L(\mathbb R^d,\mathbb R^d)$ is that, for any $i,j=1,\dots,d,$ the maps $P^{ij}x\in M\to \langle e_i,P_x e_j\rangle\in\mathbb R$ are smooth. Here $\langle\cdot,\cdot\rangle$ is the Euclidean scalar product on $\mathbb R^d.$

Working locally we can take a local orthonormal frame $V_1,\dots,V_k$ on $M.$ Then we have $$P_xe_j=e_j-\sum_{l=1}^k\langle e_j,V_l(x)\rangle V_l(x),$$ and so $$P^{ij}_x=\delta_{ij}-\sum_{l=1}^k\langle e_j,V_l(x)\rangle\langle e_i,V_l(x)\rangle$$ which is effectively a smooth functon of $x\in M.$

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