Generalised Potential Functions in Differential Geometry and Information Geometry

TL;DR

This paper investigates the applicability of potential functions to generate geometric structures in Riemannian manifolds, specifically examining their limitations with rank-four tensors from both inverse and intrinsic perspectives.

Contribution

It demonstrates that potential functions cannot generate rank-four tensors, providing new insights into the limitations of potential-based geometric constructions.

Findings

Potential functions cannot generate rank-four tensors.

Analysis from inverse and intrinsic perspectives.

Clarifies limitations of potential functions in differential geometry.

Abstract

Potential functions can be used as generating potentials of relevant geometric structures for a Riemannian manifold such as the Riemannian metric and affine connections. We study wether this procedure can also be applied to tensors of rank four and find a negative answer. We study this from the perspective of solving the inverse problem and also from an intrinsic point of view.