Directed Riemannian manifolds of pointwise constant relative sectional curvature

TL;DR

This paper introduces and characterizes a new class of Riemannian manifolds with pointwise constant relative sectional curvature, focusing on their geometric properties and specific examples like rotational hypersurfaces.

Contribution

It defines directed Riemannian manifolds of pointwise constant relative sectional curvature and provides tensor characterizations, including classification of rotational hypersurfaces and structural theorems.

Findings

All rotational hypersurfaces are directed manifolds.

Rotational hypersurfaces of pointwise constant relative sectional curvature are characterized.

Structural theorems for manifolds with totally umbilical scalar distributions.

Abstract

We study a class of Riemannian manifolds with respect to the covariant derivative of their curvature tensors. We introduce geometrically the class of directed Riemannian manifolds of pointwise constant relative sectional curvature and give a tensor characterization for such manifolds. We prove that all rotational hypersurfaces are directed and find the rotational hypersurfaces of pointwise constant relative sectional curvature. For the class of directed Riemannian manifolds of pointwise constant relative sectional curvature having a totally umbilical scalar distribution we prove a structural theorem and a theorem of Schur's type.