On parallel and symmetric 2-tensorfields on cones over pseudo-Riemannian manifolds

TL;DR

This paper classifies complete pseudo-Riemannian manifolds with cones admitting parallel symmetric 2-tensor fields, covering nilpotent, decomposable, and complex Riemannian cases, and provides explicit examples especially in the nilpotent case.

Contribution

It offers a comprehensive classification of such manifolds in complex Riemannian and decomposable cases, and describes a dense subset in the nilpotent case with examples.

Findings

Classification in complex Riemannian case

Classification in decomposable case

Examples with non-constant curvature in nilpotent case

Abstract

In this article, we study complete pseudo-Riemannian manifolds whose cone admits a parallel symmetric 2-tensorfield. The situation splits in three cases: nilpotent, decomposable or complex Riemannian. In the complex Riemannian and decomposable cases we provide a classification. In the nilpotent case, we are able to describe completely only a dense open subset of the manifold. To conclude, we give examples with non-constant curvature in the nilpotent case.