Given a Riemannian manifold \((M,g)\). We can use \(g\) to define an inner product on the cotangent bundle \(T^*M\) by raising indices
\begin{equation*} \langle \omega, \eta\rangle_g:=\langle \omega^\sharp, \eta^\sharp\rangle_g. \end{equation*}Usually we omit the \(g\) in the index.
Remark
- This definition can be used to define an inner product on an arbitrary covariant tensor space.