Curvature equations coupling symmetric tensors with a metric

TL;DR

This paper develops hierarchies of curvature equations coupling metrics with trace-free tensors, generalizing classical geometric conditions and relating to physical theories like Einstein-Maxwell and supergravity.

Contribution

It introduces a unified formalism for curvature equations involving symmetric tensors and metrics, extending classical geometric hierarchies and linking to physical models.

Findings

Derived new curvature equations for symmetric tensors and metrics.

Provided examples from geometric and algebraic constructions.

Established scalar curvature constraints generalizing classical results.

Abstract

There are described hierarchies of equations coupling a metric with a trace-free tensor having prescribed symmetries and in the kernel of certain generalized gradients. These specialize, when the tensor vanishes identically, to the usual hierarchy of constant sectional curvature (projectively flat), Einstein, and constant scalar curvature. At the Ricci curvature level these equations are formal analogues of the Einstein-Maxwell and supergravity equations that couple differential forms with a metric. The particular cases coupling a metric with trace-free symmetric tensors satisfying the Codazzi or conformal Killing equations are studied in detail. Examples of solutions are obtained from mean curvature zero immersions, affine spheres, isoparametric hypersurfaces, and related algebraic constructions. The formalism yields a hierarchy of curvature equations for statistical structures. There are deduced constraints on the scalar curvature of the metric occurring in a solution that generalize classical results of Simons, for mean curvature zero hypersurfaces in spheres, and of Calabi, for hyperbolic affine spheres.