Geometrical representations of equiaffine curvature operators

TL;DR

This paper investigates how certain algebraic curvature operators, specifically Ricci flat ones, can be represented geometrically through Ricci flat torsion-free connections on smooth manifolds.

Contribution

It proves that every Ricci flat algebraic curvature operator can be realized geometrically by a Ricci flat torsion-free connection.

Findings

Every Ricci flat algebraic curvature operator is geometrically realizable.

Provides conditions for geometric representability of equiaffine curvature operators.

Advances understanding of the link between algebraic and geometric curvature structures.

Abstract

We examine geometric representability results for various classes of equiaffine curvature operators. We show every Ricci flat algebraic curvature operator is geometrically realizable by a Ricci flat torsion free connection on the tangent bundle of some smooth manifold.