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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.

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It is just definitions. Compactness has no relation to tangent spaces. (Note, however, that the tangent space of a point is $0$-dimensional, hence is compact.) –  Zhen Lin Mar 6 '13 at 12:53

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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 $\mathbb{R}^m$.

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If a manifold is projective, such as $\mathbb{R}P^n$, then I take it the tangent space is $\mathbb{R}^n$ or would it be $\mathbb{R}^{n+1}$? –  Hal Swyers Mar 6 '13 at 13:07
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$\mathbb{R}P^n$ is an $n$-dimensional manifold and so its tangent space at any point can be modelled by $\mathbb{R}^n$ even though typically you define $\mathbb{R}P^n$ as the space of lines in $\mathbb{R}^{n+1}$. –  muzzlator Mar 6 '13 at 13:15

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