Tangent spaces of compact spaces

Question

In a recent discussion of tangent spaces, it was noted that tangent spaces to a manifold are not compact because by definition they are vector spaces. I was curious as to whether tangent spaces to compact manifolds are always non-compact. It would seem to be the case, but this appears to be more due to definitions, so I am looking for a good explanation about how compactness relates to tangent spaces.

Answer 1

Any tangent space of an $n$ dimensional manifold can be thought of as $\mathbb{R}^n$. A subset of $\mathbb{R}^n$ is compact if and only if it is closed and bounded. The space $\mathbb{R}^n$ itself is not bounded and thus it is not compact.

No matter how compact a manifold is, there are still just as many tangent vectors at a given point. When the manifold is a Lie group, compactness of the manifold can have certain implications for the Lie algebra. For example, an Ad-invariant inner product exists on the Lie algebra if and only if the Lie group is the product of a compact Lie group with $\mathbb{R}^m$.