Let $X$ be a vector field on a Riemannain manifold $M$. Consider $\nabla X:TM \to TM$, where $\nabla$ is the Levi-Civita connection of $M$.
We know that $\operatorname{tr}(\nabla X)=\operatorname{div} X$ is the divergence of $X$, which has a geometric interpretation (it determines where the flow of $X$ is volume increasing or decreasing).
I wonder: Does $\det(\nabla X)$ has a geometric interpretation as well?
I am particularly interested to know if there is any interpretation for $\nabla X$ being orientation-preserving or orientation-reversing (which corresponds to $\det(\nabla X)$ being everywhere positive or everywhere negative). Thus, even if there is no clear (quantitative) interpretation for the exact function $\det(\nabla X)$, I would like to know if there is any interpretation for it having a definite sign.
Note that $\nabla X:TM \to TM$ is (fiberwise) a self-map of a vector space, so its determinant is well-defined, even without any choice of orientation on $M$. (Though we may as well choose an orientation locally, or even fiberwise, so this does not really matter).