a question about compact tangent bundle

Question

I have a question about tangent bundles. Is there a compact tangent bundle? Or what conditions do we need to be sure that tangent bundle of a manifold be compact?

Answer 1

The tangent bundle to a non trivial manifold is never compact, for the fiber over a point $m\in M$ under the continuous projection $$\pi:TM\to M$$ is a closed subspace (equal to $T_mM$) of $TM$, yet it is never compact, for diffeomorphic to $\Bbb R^n$.

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Of course the tangent bundle of a $0$-dimensional manifold (i.e. a discrete space) is equal to the manifold itself, so finite discrete spaces are manifolds with compact tangent bundles as it were.