what is the $p^{*}TM$ in Chern's book : Lectures on Differential Geometry

Question

In Chern's book, $PTM$ is the projectivised tangent boundle of m-dimentional manifold $M$, if $p:PTM \rightarrow M$ is the pulled back map, then he says $p^{*}TM$ is the vector boundle with the base manifold $PTM$, and its fibers are vector spaces of dimension m. So how to understand this definition, and what is the difference between $p^*TM$ and $p^*T^*M$.

Answer 1

The general setup is this: Suppose we have two topological spaces $M$ and $N$. Suppose that $F$ is a bundle over $N$ and that $\phi: M \to N$ is a continuous function. There is a pullback bundle over $M$ denoted $\phi^*F$ whose fiber at $p \in M$ is $F_{\phi(p)}$. You can learn more about the definition on Wikipedia. They are also covered in (for example) Milnor and Stasheff's Characteristic Classes.

In this case, one has a manifold $M$ with projectivised tangent bundle $PTM$. There is a natural projection map $p: PTM \to M$. As the total space of $PTM$ is a topological space, the bundle $TM$ over $M$ pulls back to $p^*TM$ over $PTM$.

$TM$ and $T^*M$ are the tangent and cotangent bundles of $M$. So $p^*TM$ and $p^*T^*M$ are the respective pullbacks.