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.

Links Link to heading

• Riemannian metric
• cotangent bundle
• musical isomorphism

See also Link to heading

• inner product of covectors in local coordinates
• inner product of tensors