Using the inner product of covectors we may define a smooth fiber metric on each tensor bundle \(T^{(k,l)}TM\). A smooth fiber metric is an inner product on each fiber \(T^{(k,l)}T_pM\) which varies smoothly.
Let \(\alpha_1,\ldots,\alpha_{k+l}\) and \(\beta_1,\ldots,\beta_{k+l}\) be vector or covector fields as appropriate, then
\begin{equation*} \langle \alpha_1\otimes \cdots \otimes \alpha_{k+l}, \beta_1\otimes \cdots \otimes \beta_{k+l}\rangle=\langle \alpha_1, \beta_1\rangle \cdot \ldots\cdot \langle \alpha_{k+l}, \beta_{k+l}\rangle, \end{equation*}then defines a unique smooth fiber metric on \(T^{(k,l)}TM\).
Remark
- The induced smooth fiber metric of two smooth functions is just the product of them.
- The induced metric is bilinear.