Riemaniann metric is an element of?

Question

We are given the definition:

A riemaniann metric $g$, is a map:

$g:p\rightarrow<.,.>|_{T_pM}$

where

$<.,.>|_{T_pM}$

is the usual bilinear symmetric etc..

It also says that the metric g is an element of $\otimes^2 T^*M$ but is not the image of g that should be regarded as an element of that space? can someone please explain

Answer 1

We normally think of $\otimes^2 T^*M$ as a vector bundle over $M$ and say that $g$ is an element of $\Gamma(M, \otimes ^2 T^*M)$, the section of the bundle. That is, for each $x\in M$, $g(x) = \langle \cdot, \cdot \rangle_x$ is an element of $\otimes^2 T^*_xM$.