I'm just starting to learn about vector bundles, I want to compute the transition functions of the bundle $TS^n$.
I started with the stereographic atlas $U_1 = S^n – \{N\}$ and $U_2 = S^n – \{S\}$ where $\phi_i : U_i \rightarrow \mathbb{R}^n$ is the diffeomorphism given by the stereographic projection.
I have computed explicitly these applications and their inverses. My question is, do these applications induce local trivializations for $TS^n$? In that case, the transition map at $x$ is given by
$$d_x(\phi_2 \circ \phi_1^{-1})$$
am I right?
Best Answer
The tangent bundle of a manifold $M$ is a vector bundle, denoted $(TM,\pi,M)$ where $\pi(p,V_p)=p$. Given coordinate chart $(U,\varphi)=(U,x^1,\dots x^n)$ on $M$ we get the induced map $$T\varphi:\pi^{-1}(U)\to U\times \mathbb{R}^n \ \ \ \ \ \ \ \ \ \ (p,V^i\frac{\partial}{\partial x^i})\mapsto (p,V^1,\dots,V^n)$$
You can verify that $\pi_1\circ T\varphi=\pi$, where $\pi_1:U\times \mathbb{R}^n\to U$ is the projection, and that for each $p\in M$, the map $T\varphi|_p:T_pM\to\{p\}\times\mathbb{R^n}$ is an isomorphism. That is, each coordinate chart $(U,\varphi)$ is a local trivialization of $TM$.
As for the transition functions if I understand your notation then what you have is correct:
Suppose that $(U,\phi_1)=(U,x^1,\dots,x^n)$ and $(V,\phi_2)=(V,y^1,\dots,y^n)$ are two coordinate charts. Then for arbitrary $(p,W^1,\dots,W^n)\in (U\cap V)\times \mathbb{R}^n$ we have that $$\phi_2\circ\phi_1^{-1}\left(p,W^1,\dots,W^n\right)=\phi_2\left(q,W^i\frac{\partial}{\partial x^i}\right)=\phi_2\left(q,W^i\frac{\partial y^j}{\partial x^i}\frac{\partial}{\partial y^j}\right)=\left(p,\frac{\partial y^1}{\partial x^i}W^i,\dots,\frac{\partial y^n}{\partial x^i}W^i\right)$$so that the transition functions are given by the Jacobian.