Generalized warped products and the $\kappa$-nullity of Riemannian curvature

TL;DR

This survey explores how certain inhomogeneous, curvature homogeneous Riemannian manifolds with nontrivial -nullity can be understood as deformations of homogeneous metrics via Riemannian submersions, and discusses open questions.

Contribution

It provides a unified perspective on known examples of -nullity manifolds as deformations of homogeneous spaces through Riemannian submersions.

Findings

Examples of -nullity manifolds are deformations of homogeneous metrics.

Connections between inhomogeneous and homogeneous Riemannian manifolds are established.

Open questions about -nullity and warped products are posed.

Abstract

In this short survey, we show how two (classes of) known examples of inhomogeneous, curvature homogeneous Riemannian manifolds with nontrivial $\kappa$-nullity can be seen as deformations of homogeneous metrics along the vertical distribution of an integrable Riemannian submersion. We also pose two open questions.