A special class of symmetric Killing 2-tensors

TL;DR

This paper investigates symmetric Killing 2-tensors on Riemannian manifolds, revealing unique properties of Sasakian manifolds and spheres, and characterizing three-Sasakian manifolds through their characteristic forms.

Contribution

It identifies conditions under which symmetric Killing 2-tensors characterize Sasakian manifolds and spheres, extending previous results and providing new characterizations.

Findings

Sasakian manifolds and spheres satisfy specific symmetric Killing 2-tensor conditions

Three-Sasakian manifolds can be characterized via properties of their characteristic 1-forms

A new proof or extension of Gallot's sphere characterization using third-order differential equations

Abstract

We study symmetric Killing 2-tensors on Riemannian manifolds and show that several additional conditions can be realised only for Sasakian manifolds and Euclidean spheres. In particular we show that (three)-Sasakian manifolds can also be characterized by properties of the symmetric products of their characteristic 1-forms. Moreover, we recover a result of S.~Gallot on the characterization of spheres by means of functions satisfying a certain differential equation of order three.