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Let $M$ be a smooth manifold and $S\subseteq M$ a properly embedded smooth submanifold. Suppose that we have a fibre metric on $TM|_S$, i.e. a positive definite real inner-product on $T_pM$ for all $p\in S$, which depends smoothly on $p\in S$. Can it always be extended to a Riemannian metric on $TM$?

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    $\begingroup$ Yes. Use bump functions to reduce to the case of $S$ a linear subspace of a vector space $M$. $\endgroup$
    – Ben McKay
    Commented Oct 4, 2017 at 14:22

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