Curvature properties and isotropic planes of Riemannian and almost Hermitian manifolds of indefinite metrics

TL;DR

This paper investigates the curvature properties of Riemannian and almost Hermitian manifolds with indefinite metrics, focusing on isotropic planes and their relation to conformal flatness and constant curvature.

Contribution

It establishes new characterizations of conformally flat and constant curvature manifolds via isotropic planes and extends these results to almost Hermitian manifolds with indefinite metrics.

Findings

Manifolds with vanishing curvature tensor on strongly isotropic planes are conformally flat.

Manifolds satisfying the plane axiom are of constant sectional curvature or conformally flat.

Analogous properties are studied for almost Hermitian manifolds with indefinite metrics.

Abstract

We study two types of isotropic planes: weakly isotropic and strongly isotropic planes. We prove that a Riemannian manifold of indefinite metric is conformally flat if and only if its curvature tensor vanishes on all the strongly isotropic planes. We specialize the plane axiom for Riemannian manifolds of indefinite metrics. We show that manifolds satisfying plane axiom of weakly (strongly) isotropic planes are of constant sectional curvature (conformaly flat). Further we study analogous problems on almost Hermitian manifolds of indefinite metrics.