A second-order identity for the Riemann tensor and applications

TL;DR

This paper derives a new second-order differential identity for the Riemann tensor on manifolds with symmetric connection, revealing new and existing identities and exploring applications to special geometric structures like recurrent and symmetric manifolds.

Contribution

It introduces a novel second-order differential identity for the Riemann tensor and uncovers a new K-recurrency structure from Lovelock's invariance, expanding geometric analysis tools.

Findings

New second-order identity for Riemann tensor

Derivation of identities for Ricci and Riemann tensors

Identification of K-recurrency structure in geometric manifolds

Abstract

A second-order differential identity for the Riemann tensor is obtained, on a manifold with symmetric connection. Several old and some new differential identities for the Riemann and Ricci tensors descend from it. Applications to manifolds with Recurrent or Symmetric structures are discussed. The new structure of K-recurrency naturally emerges from an invariance property of an old identity by Lovelock.