Let \(u\) be a smooth real-valued function on a [smooth manifold](smooth manifold.md) \(M\). Then the second covariant derivative \(\nabla^2u\) is called covariant Hessian of \(u\).
The local expression of \(\nabla^2u\) is given by
\begin{equation*} \nabla^2u=u_{;ij}dx^i\otimes dx^j, \qquad \text{with } u_{;ij}=\partial_j\partial_iu-\Gamma^k_{ji}\partial_ku, \end{equation*}where \(\Gamma^k_{ji}\) denotes the connection coefficients of \(\nabla\).
We also may write \(u_{i_1\ldots i_k}\) as an abbreviation of \(u_{;i_1\ldots i_k}\).
Remarks
- The covariant Hessian is a smooth covariant 2-tensor field.
- The convention is that \(u\) is first differentiated by the first index on the left.
- For the Levi-Civita connection \(\nabla^2 u\) is symmetric.