Geometric theory of equiaffine curvature tensors

TL;DR

This paper develops an algebraic framework for equiaffine curvature tensors, exploring their decompositions and applications to geometric structures like Codazzi structures and projectively flat hypersurfaces.

Contribution

It introduces new algebraic decompositions of equiaffine curvature tensors and applies these to characterize various geometric structures.

Findings

Decomposition of equiaffine curvature tensors into orthogonal irreducible components

Characterization of Codazzi structures using tensor decompositions

Analysis of projectively flat structures in the context of equiaffine geometry

Abstract

We present an algebraic investigation of generalized and equiaffine curvature tensors in a given pseudo-Euclidean vector space and study different orthogonal, irreducible decompositions in analogy to the known decomposition of algebraic curvature tensors. We apply the decomposition results to characterize geometric properties of Codazzi structures and relative hypersurfaces; particular emphasis is on projectively flat structures.