What does $TM$, $T^*M$ and $*$ mean?(Definition Check)

Question

I was reading Hamilton's Ricci Flow by Bennett Chow Peng Lu and Lei Ni.

However, I encountered some notation confusion:

1. What does $TM$ mean? Does $T$ stands for tensor or tangent, "M" for metric?

2. What's the difference between $TM$ and $T^*M$?

3. Also, there seemed to be a custom to use $*$ for either space or functions, i.e. $V^*$ or $\varphi_*$. What does $*$ indicate here? Does it indicate dual space or something else?

(The preview of the material is available on amazon. )

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Answer 1

$T_pM$ is the space of vectors tangent to $M$ at $p$, and $T_p^*M=(T_pM)^*$ is the corresponding dual space.

$TM$ is the tangent bundle of manifold, $T^*M$ is the cotangent bundle of the manifold. $$ TM=\bigsqcup_{p\in M} T_pM$$ subject to the "gluing" conditions that $TM$ is "locally" of the form $U\times \mathbb{R}^n$ for $U$ an open neighborhood of $M$. These objects are called smooth vector bundles and you can read more about them in a book like

$(1)$ Lee's Introduction to Smooth Manifolds

$(2)$ Tu's An Introduction to Manifolds.