Geometric realizations of curvature

TL;DR

This paper explores how various types of curvature in different geometric structures can be realized and characterized, connecting classical and modern geometric concepts.

Contribution

It provides a unified framework for understanding geometric realization problems across multiple curvature-related geometries.

Findings

Characterization of curvature realizations in diverse geometric settings

Connections established between Ivanov-Petrova and Osserman geometries

Insights into curvature homogeneity across different structures

Abstract

We study geometric realization questions of curvature in the affine, Riemannian, almost Hermitian, almost para Hermitian, almost hyper Hermitian, almost hyper para Hermitian, Hermitian, and para Hermitian settings. We also express questions in Ivanov-Petrova geometry, Osserman geometry, and curvature homogeneity in terms of geometric realizations.