Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $s$. Also, 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 $s$ has a modulus of continuity.

share|cite|improve this question
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
up vote 3 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 $x\in M.$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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