Vector Fields as a section of the Tangent bundle

Question

In most textbooks for differential geometry a vector field on a smooth manifold $M$ is said to be a section of the corresponding tangent bundle $TM$. I give the definition I am provided with and then explain my confusion about this definition:
A smooth vector field on $M$ is a smooth map \begin{align} X:M\rightarrow TM \end{align} such that \begin{align} \pi\circ X = id_M\end{align} where $\pi:TM\ni (p,v_p)\rightarrow p\in M$ is the tangent bundle projection.
In my lecture notes it says, that the last definition ensures that $X(p) \in T_pM$. This is where my confusion starts. Is $X(p)$ now an element of the tangent bundle $TM$ as in the definition above or is it an element of the tangent space $T_pM$? Shouldn't one cosider the map $v:M \rightarrow T_pM$ defined in \begin{align} X: M\ni p \rightarrow (p,v(p)) \in TM \end{align} as the vector field instead of $X$?

Answer 1

Intuitively, a vector field associates, with each point $p$ on the manifold, a vector in $T_pM$. And we would like to phrase this association as a function. Then we have the challenge: What is the codomain of this function? Clearly it can't be just $T_pM$, because what about the vector field values at points other than $p$?

You say $v:M\to T_pM$, but this runs into exactly this problem. Where does $v(q)$ live, for $q\neq p$? We would like it to live in $T_qM$, but by our definition it lives in $T_pM$, which is a completely different space.

The most sensible codomain for the entire vector field is the full tangent bundle $TM$. As a set, the tangent bundle is the union of all the different tangent spaces $T_pM$ at all possible points $p\in M$, so it's in a natural way the smallest possible codomain we could hope for. Then, of course, since we want the vector field's value at $p$ actually does lie in $T_pM$ (rather than some other $T_qM$), we have to add the extra stipulation that $\pi\circ X = id_M$.

Answer 2

One often thinks of the tangent space $T_p M$ as subset of the tangent bundle $TM$ and in this sense you can understand the statement $X(p)\in T_pM$.

I believe that part of your confusion stems from the notation $(p,v)$ for elements in $TM$. As a set $TM$ can be viewed as a disjoint union of all tangent spaces $TM = \coprod_{p\in M} T_p M$ and the projection $\pi\colon TM\to M$ simply tells you in which tangent space an element of $TM$ lives. For this reason some people denote elements in $TM$ simply by $v$, keeping in mind that $v\in T_p M$, where $p:=\pi(v)$. In this sense it is redundant to include $p$ in the notation for elements of $TM$. Another group of people likes to include $p$ in the notation anyways (say, to avoid always having to define $p:=\pi(v)$) and instead of $v$ they prefer the notation $ (p,v) $ for elements in $TM$. Writing this as a tuple is merely a notational device and you should keep in mind that $TM$ is not (in any natural way) a Cartesian product of two sets.