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?