The Laplacian of \(u\) may be defined using the covariant Hessian \(\nabla^2u\). The following identity holds
\begin{equation*} \Delta u = \tr_g (\nabla^2u), \end{equation*}where \(\tr_g\) denotes the trace operator and \(\nabla u\).
Proof
This is an application of (0x66deed94)
(trace representation of divergence), (0x66deb7b0)
(\(\sharp\) commutes with \(\nabla\)) and the definition of \(\nabla^2\) as wells as \(\tr_g\).