Generalized Double Duals of the Riemann Tensor in Geometry and Gravity

TL;DR

This work addresses the limitation that Ricci contractions of the Riemann tensor discard Weyl information by introducing a canonical hierarchy of divergence-free, symmetric $(p,p)$ double forms ${}^{\ast}{\mathbf{R}}^{\ast}_p$ obtained from the double dual of the Riemann tensor. The authors establish index-free and componentwise descriptions, prove uniqueness theorems for these higher-rank tensors under generic geometric conditions, and relate their $p$-curvatures $s_p$ to sectional curvatures, showing that $s_2$ determines the full Riemann tensor and yields topological consequences such as vanishing of the $\hat{A}$-genus under $s_2\ge 0$. They then extend the construction to Gauss–Kronecker curvature tensors and Lovelock theory via $(p,q)$-curvature tensors, revealing a hierarchy of higher-index parents of Lovelock tensors and providing a four-index gravity perspective through contractions involving ${}^{\ast}{\mathbf{R}}^{\ast}_2$. The framework unifies four-index gravity proposals with Lovelock gravity, clarifies how curvature information propagates through contractions, and suggests modifications of gravitational field equations that preserve richer geometric data. Overall, the paper offers a rigorous, canonical way to access full curvature information with higher-rank, divergence-free tensors and demonstrates substantial geometric and physical implications, including positivity results, topological constraints, and a broader gravitational formalism.

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.

Theorems & Definitions (42)

• Definition 2.1
• Proposition 2.1
• proof
• Corollary 2.2
• proof
• Corollary 2.3
• proof
• Proposition 2.4
• proof
• Definition 2.2