Geometric realizations of generalized algebraic curvature operators

TL;DR

This paper investigates the geometric realization of generalized algebraic curvature tensors, addressing eight GL-equivariant questions, and finds that all but one are solvable, with a specific exception related to projectively flat Ricci antisymmetric cases.

Contribution

It provides a comprehensive analysis of geometric realization questions for generalized algebraic curvature tensors, identifying which cases are solvable and highlighting a notable exception.

Findings

All but one of the eight geometric realization questions are solvable.

A non-zero projectively flat Ricci antisymmetric tensor cannot be realized by a torsion-free connection.

The study clarifies the limitations and possibilities in realizing algebraic curvature structures geometrically.

Abstract

We study the 8 natural GL equivariant geometric realization questions for the space of generalized algebraic curvature tensors. All but one of them is solvable; a non-zero projectively flat Ricci antisymmetric generalized algebraic curvature is not geometrically realizable by a projectively flat Ricci antisymmetric torsion free connection.