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I have a basic question about vector fields on Riemannian manfiold: Suppose we have a smooth vector field $\xi $ sending $ x \to \xi_x \in T_x \mathcal{X}$ defined on a compact subset $\mathcal{X}$ of the manifold? Can one assume $\|\xi_x\| \leq C$ for some constant $C$ unifolmly for all $x \in \mathcal{M}$?

Is there a proof of this or a counterexample?

user143234
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    Let $K\subset \mathcal X$ be the compact subset on which $\xi$ is defined. Then $|\xi|:K\to \mathbb R$ is a real-valued function on a compact topological space. – Danu Sep 01 '20 at 10:03
  • Thanks, i knew it was something simple . – user143234 Sep 01 '20 at 10:06

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