Generalized Double Duals of the Riemann Tensor in Geometry and Gravity

TL;DR

This paper introduces a hierarchy of divergence-free tensors derived from the double dual of the Riemann tensor, revealing new geometric invariants and extending to Lovelock gravity, with implications for curvature and topology.

Contribution

It constructs canonical higher-rank divergence-free tensors from the Riemann tensor's double dual, generalizing Einstein tensors and linking curvature to topological invariants.

Findings

The hierarchy includes unique divergence-free tensors depending linearly on Riemann curvature.

The 2-curvature tensor determines the full Riemann tensor and affects the $ ext{A}$-genus.

Extension to Lovelock tensors provides new insights into higher-order curvature theories.

Abstract

The Riemann curvature tensor fully encodes local geometry, but its Ricci contraction retains only limited information: only the Ricci tensor and the scalar curvature survive, while the Weyl curvature vanishes identically. We show that contracting instead the double dual of the Riemann tensor unlocks the full curvature structure, producing a canonical hierarchy of symmetric, divergence--free $(p,p)$ double forms. These tensors satisfy the first Bianchi identity and obey a hereditary contraction relation interpolating between the double dual tensor and the Einstein tensor. We prove that, in a generic geometric setting, each tensor in this hierarchy is the unique divergence--free $(p,p)$ double form depending linearly on the Riemann curvature tensor, thereby providing canonical higher--rank parents of the Einstein tensor. Their sectional curvatures coincide with the $p$--curvatures; notably, the $2$--curvature determines the full Riemann curvature tensor and forces the $\hat A$--genus of a compact spin manifold to vanish when nonnegative, a property not shared by Ricci or scalar curvature. Finally, we extend the construction to Gauss--Kronecker curvature tensors and Lovelock theory, showing in particular that the second Lovelock tensor $T_4$ admits a genuine four--index parent tensor.