Let $(M,\langle \cdot,\cdot\rangle)$ be a pseudo-Riemannian manifold and $f: M \to \Bbb R$ be a smooth function. One can consider the covariant Hessian $\nabla ({\rm d}f)$. Some time ago I had seen a paper, which regrettably I do not recall the name, studying the pseudo-Riemannian geometry of $M$ equipped with $\nabla({\rm d}f)$ (when said Hessian is non-degenerate). In particular, the author(s) obtain an (coordinate-free) expression for the Levi-Civita connection of the Hessian metric in terms of the connection of the original metric.

The Pacific Journal of Mathematics and the Balkan Journal of Geometry and Its Applications both come to mind, but looking through these websites I did not found anything.

Maybe someone here happens to know the reference off the top of their head. If this question seems off-topic, I'll just delete it. Thanks!


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