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Let $X$ be a complex manifold equipped with a smooth hermitian metric $h$. We can define a sub-fibration $B \to X$ of the tangent bundle $T_X$ by requiring that the fiber over a point be the unit ball in $T_X$ at that point, i.e.

$ B_x = \{ \xi \in T_{X,x} \, | \, \| \xi \|_{h(x)} = 1 \}. $

I want to see that if $X$ is compact, then the fibration $B$ is compact in the total space of $T_X$. This is just a global version of the usual fact that the unit ball in a normed finite dimensional vector space is compact.

I can prove this by using the (horrible) definition of $T_X$ as the disjoint union of the stalks $T_{X,x}$ modulo an equivalence relation. As each $B_x$ is compact, their disjoint union is compact in the disjoint union of the stalks by Tychonoff, and dividing by the equivalence relation is a continuous map so $B$ is compact in the total space of $T_X$.

Isn't there a nicer way to see this? I ask both because I really don't like the definition of the tangent space as a disjoint union of stalks, and because I have to prove a similar lemma for the relative tangent space associated to deformations of a manifold $X$. A similar proof works in that case, but it is quite dirty.

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    It is, but to make the trick work one would have to deduce that from that $\mathbb P^n$ and $S^1$ are compact.2011-02-23

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Hint: Here is an outline of a possible proof.

1) Do the problem in the special case that the ball fibration is trivial, i.e., isomorphic to a product.

2) Convince yourself that the ball fibration is locally trivial.

3) Since the base is compact, you can find a finite open cover $\{U_i\}$ such that the restriction of the fibration to each $U_i$ is trivial. Now use the fact that a finite union of compact sets is compact.

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    Pete's solution is not exactly what I want because it is morally the same as the proof I outlined in my question. Like I said, I can make that one work for what I need, but I wondered if there wasn't a trick or a clever idea one could use to make this trivial. In short, isn't there a Book proof of this?2011-02-23