Riemannian geometric realizations for Ricci tensors of generalized algebraic curvature operators

TL;DR

This paper investigates the geometric realizability of algebraic curvature operators in affine and Riemannian geometry, constructing connections with prescribed curvature properties and constant scalar curvature.

Contribution

It provides methods to realize algebraic curvature operators as curvature tensors of torsion-free connections with specific Ricci properties.

Findings

Constructs torsion-free connections with given algebraic curvature operators.

Ensures connections have constant scalar curvature.

Realizes Ricci symmetric, trace-free Ricci, and Ricci alternating conditions.

Abstract

We examine questions of geometric realizability for algebraic structures which arise naturally in affine and Riemannian geometry. Suppose given an algebraic curvature operator R at a point P of a manifold M and suppose given a real analytic (resp. C-k for finite k at least 2) pseudo-Riemannian metric on M defined near P. We construct a torsion free real analytic (resp. C-k) connection D which is defined near P on the tangent bundle of M whose curvature operator is the given operator R at P and so that D has constant scalar curvature. We show that if R is Ricci symmetric, then D can be chosen to be Ricci symmetric; if R has trace free Ricci tensor, then D can be chosen to have trace free Ricci tensor; if R is Ricci alternating, then D can be chosen to be Ricci alternating.