The divergence may be defined with a Levi-Civita connection . In this case we have
\begin{equation*} \div X = \tr(\nabla X). \end{equation*}[1, Problem 5.14]
Proof
To proof this equality consider for each point \(p\) normal coordinates
. Since \(g_{ij}=\delta_{ij}\) and \(\Gamma_{ij}^k=0\) in \(p\) the identity is easy to prove using local representation of the divergence
and the one of total covariant derivatives
. Since equality holds for every point, we are done.
Remarks
- In local expression the divergence of \(X\) is given by \(\div X = {X^i}_{;i}\).
References Link to heading
- J. Lee, Introduction to Smooth Manifolds. New York ; London: Springer, 2013.