On some classes of Riemannian manifolds

TL;DR

This paper investigates specific classes of Riemannian manifolds defined by Ricci tensor conditions, transforming the associated PDE systems into involutive forms to find explicit solutions and explore their interrelations.

Contribution

It provides a systematic approach to classify and explicitly construct examples of manifolds satisfying Ricci-related conditions using involutive PDE systems.

Findings

Explicit families of nontrivial manifolds satisfying Ricci conditions

Transformation of PDE systems into involutive forms for analysis

Relationships between different Ricci-related classes

Abstract

We study several classes of Riemannian manifolds which are defined by imposing a certain condition on the Ricci tensor. We consider the following cases: Ricci recurrent, Cotton, quasi Einstein and pseudo Ricci symmetric condition. Such conditions can be interpreted as overdetermined PDE systems whose unknowns are the components of the Riemannian metric, and perhaps in addition some auxiliary functions. Hence even if the dimension of the manifold is small it is not easy to compute interesting examples by hand, and indeed very few examples appear in the literature. We will present large families of nontrivial examples of such manifolds. The relevant PDE systems are first transformed to an involutive form. After that in many cases one can actually solve the resulting system explicitly. However, the involutive form itself already gives a lot of information about the possible solutions to the given problem. We will also discuss some relationships between the relevant classes.