This question was asked in my quiz of Differential geometry course and I am having a really hard time in this course.
Question: Let $M$ be an oriented manifold of dimension $n$ and let $π : TM \to M$ be it's tangent bundle.
(i) What are the transition functions of $TM?$
(ii) What are the transition functions of the bundle $T^* M \otimes_M T^* M \to M?$
I have followed my lecture notes and I don't quite understand exactly how to define transition functions here?
I am also reading textbook of Frank Warner but I don't think he covers transition functions.
Can you please help me with answering this question?
I'll address (i), which appears to ask which transition functions preserve an orientation.
Suppose we have two overlapping charts $(U, \phi)$, $(V, \psi)$ on $M$. Both $\phi(U), \psi(V)$ inherit from $\Bbb R^n$ the standard volume form $\operatorname{vol}$, which pull back to volume forms (hence determine orientations on $U, V$). The transition function $$\tau := \psi^{-1} \circ \phi$$ pulls back $\operatorname{vol}$ to $$({\det T\tau}) \operatorname{vol},$$ where $T\tau$ is the tangent map of $\tau$. So $\tau$ preserves orientation---equivalently, $\phi$ and $\psi$ determine the same orientation on $U \cap V$---iff $\det T\tau > 0$.