I noticed several times online that people tend to use the symbol $\Gamma(M,TM)$ when talking about the space of smooth vector fields on smooth manifolds. I find this totally confusing, as in differential geometry this should be the same as $C^{\infty}(M,TM).$ Thus, I don't see why one would introduce this strange notation. Can anbody motivate it or is there even a difference between the two spaces?
2 Answers
Often one uses $\Gamma$ to denote the "global sections" of a sheaf. $TM$ is a vector bundle (so a locally free sheaf) whose global sections are the smooth vector fields on $M$. This is the reason the set of all smooth vector fields on $M$ is denoted by $\Gamma(M,TM)$.
To be explicit, by global section we mean a smooth map $f:M\to TM$ such that $\pi\circ f = \text{id}$ where $\pi:TM\to M$ is the projection (here global means defined on all of $M$ rather than just some open subset).
$C^\infty(M,TM)$ is the set of all smooth functions from $M$ to $TM$, but these functions need not be sections.
A smooth (real) vector bundle of rank $k\in\mathbb N$ is a triple $(E, \pi, B)$ consisting of smooth manifolds $E$, $B$ and a surjective smooth map $\pi:E\longrightarrow B$ such that:
$(i)$ $E_b:=\pi^{-1}(b)$ is a vector space over $\mathbb R$;
$(ii)$ For every $p\in B$ there exists an open subset $U\subseteq B$, $p\in U$, and a diffeomorphism $\phi:\pi^{-1}(U)\longrightarrow U\times \mathbb R^k$ such that $\textrm{pr}_1\circ \phi=\pi$.
$(iii)$ $\phi|_{E_p}:E_p\longrightarrow \{p\}\times \mathbb R^k$ is a linear isomorphism for every $p\in B$.
$E$ is called the total space, $B$ the base, $\pi$ the projection and $E_b$ the fiber over $b$.
You may see the tangent bundle of a smooth manifold is naturally a vector bundle.
Whenever you have a vector bundle $(E, \pi, B)$ you may define $$\Gamma(E):=\{f:B\longrightarrow E; f\ \textrm{is smooth and}\ \pi\circ f=1_B\}.$$
An element $f\in \Gamma(E)$ is called a smooth section of $E$.
Notice $f\in \Gamma(E)$ if and only if $f(b)\in E_b$ for every $b\in B$.
$\Gamma(E)$ is an $\mathbb R$-vector space (indeed it has other algebraic structures).
Now you may recognize a smooth vector field as a section of the bundle $(TM, \pi, M)$ where $\pi:TM\longrightarrow M$ is the projection which assigns the base point of a tangent vector on $M$.
You may check Introduction to smooth manifolds (J. Lee) for a nice introduction to vector bundles.
As to the notation, it's pretty common using $\Gamma(E)$ to denote the space of smooth sections of a bundle but in some papers other notations also appear.
