The Levi-Civita connection induces a connection on every tensor bundle. It is compatible with the induced inner product, i.e. for \(X\in \mathfrak{X}(M)\) and \(F,G\in \Gamma(T^{(k,l)}TM)\)

\begin{equation*} X\langle F, G\rangle=\langle \nabla_XF, G\rangle+\langle F, \nabla_XG\rangle . \end{equation*}

(For more informations see [1].)

Links Link to heading

• Levi-Civita connection
• induced connections on tensor bundles
• inner product on tensor bundles

References Link to heading

1. J. Lee, Introduction to Smooth Manifolds. New York ; London: Springer, 2013.