Curvature equations coupling symmetric tensors with a metric
Daniel J. F. Fox
TL;DR
The paper develops a general scheme for coupling a pseudo-Riemannian metric $h$ with a trace-free tensor $\omega$ of prescribed Young-diagram symmetry, via generalized Stein–Weiss gradients, to produce a hierarchy of curvature equations (top-level projective-flat and Einstein, plus constraint levels). By selecting $\omega$ to satisfy linear generalized gradient equations and choosing a quadratic map $\phi$ (e.g., involving $\omega\wedge\omega$), the curvature of $h$ is tied to quadratic expressions in $\omega$, extending the classical hierarchies for constant curvature, Einstein, and constant scalar curvature. The paper provides concrete classes of solutions from submanifold geometry (mean curvature-zero immersions, Legendrian immersions), statistical structures, invariant polynomials on Lie algebras, and other algebraic constructions, and develops Weitzenböck formulas, refined Kato inequalities, and Calabi–Cheng–Yau–type estimates to constrain the growth and pinching of $\omega$, yielding scalar-curvature and Simons-type integral bounds. Together, these results establish a robust framework linking differential geometry, submanifold theory, and algebraic constructions to new curvature-coupled systems with potential links to physics-inspired equations such as Einstein–Maxwell and supergravity-type couplings. The significance lies in a unified, representation-theoretic approach to curvature–tensor couplings that produces nontrivial high-dimensional examples and rigorous constraints on solution behavior.
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.