Let $M$ to be a manifold $m$-dimensional with a smooth hermitian metric $g$. The tangent bundle of $M$ is given by $TM= \bigcup_{p\in M} T_{p}M$, and the unit tangent bundle is given by
$S=\{x \in TM: \|x\|_{g(p)} =1$ where $x \in T_{p}M\} $.
(1)
In If M is a compact manifold, how to prove that the unitary tangent bundle is also compact?, the solutions assumed that given $x$ in $M$, you can obtain $U$ open neighborhood of $x$ such that $\pi^{-1}(U) \cap S \cong U \times S^{n-1}$. But this step is very important, and they do not show this homeomorphism.
(2)
In "The unit ball fibration in a tangent bundle", Pete L. Clark showed that $S$ is a compact set of $TM$ if $M$ is a compact manifold $m$-dimensional:
"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."
I was trying to understand this answer, but I failed to follow this steps.
First of all, we consider $\pi: TM \to M$ the canonical projection. Using that $M$ is a compact manifold, there exists a finite charts $(\phi_{i}, U_{i})_{i=1}^{n}$ that covers $M$.
We note that if $M$ is a compact manifold then $TM$ is a trivial bundle, just consider $\Phi: TM \to M \times \mathbb{R}^{m}$ given by $x \mapsto (\pi(x), \theta_{\pi(x)}^{(U_{i}, \varphi_{i})} (x)) $ where $\theta_{\pi(x)}^{(U_{i}, \varphi_{i})}: T_{\pi(x)} M \to \mathbb{R}^{m}$ is a isomorphism linear and $i = \min\{j : \pi(x) \in U_{j}\}$.
But I'm stuck in this part. Can some one give some hint?
Thank you
Recall the definition:
A vector bundle of rank $m$ over $M$ is a topological space $E$ together with a surjective continuous map $\pi:E \to M$ satisfying the following conditions:
($1$) For each $p \in M$, the fiber $E_{p}= \pi^{-1}(p)$ over $p$ is endowed with the structure of a $k$-dimensional real vector space.
($2$) For each $p$ in $M$, there exists a neighborhood U of p in M and a homeomorphism $\Phi: \pi^{-1}(U) \to U \times \mathbb{R}^{m}$ (called a local trivialization of $E$ over $U$), satisfying the following conditions:
(2.1) $\pi_{U} \circ \Phi = \pi$ (where $\pi_{U}:U\times \mathbb{R}^{m} \to U$ is the projection);
(2.2) for each $q$ in $U$, the restriction of $\Phi$ to $E_{q}$ is a vector space isomorphism from $E_{q}$ to $\{q\}\times\mathbb{R}^{m}$.