Let $(M,g)$ be a Riemannnian manifold and let $f:\Sigma\to M$ be a smooth immersion. Then the vector bundle $f^\ast TM\to\Sigma$ has a natural bundle metric and metric-compatible connection. Can one characterize the situations in which there must exist a section $V$ such that $\nabla V=df$? This is trivially possible if $(M,g)$ is Euclidean space. It feels like it should not be possible in general.

This seems to be equivalent to the existence of a closed 1-form $\omega$ on $\Sigma$ and a normal vector field $w$ along $\Sigma$ such that \begin{align}\nabla \omega-\langle h,w\rangle&=f^\ast g\\ h(\cdot,\omega^\sharp)+\nabla^\perp w&=0\end{align} where $h$ is the second fundamental form. I can't see any immediate conclusions to make.