Clifford structures on Riemannian manifolds

TL;DR

This paper introduces even Clifford structures on Riemannian manifolds, providing a comprehensive classification of manifolds with such structures, including various symmetric spaces and special geometries.

Contribution

It generalizes existing geometric frameworks and classifies all manifolds with parallel even Clifford structures, extending known results in related geometries.

Findings

Complete classification of manifolds with parallel even Clifford structures

Identification of symmetric spaces associated with these structures

Extension of results in Sasakian and 3-Sasakian geometry

Abstract

We introduce the notion of even Clifford structures on Riemannian manifolds, a framework generalizing almost Hermitian and quaternion-Hermitian geometries. We give the complete classification of manifolds carrying parallel even Clifford structures: K\"ahler, quaternion-K\"ahler and Riemannian products of quaternion-K\"ahler manifolds, several classes of 8-dimensional manifolds, families of real, complex and quaternionic Grassmannians, as well as Rosenfeld's elliptic projective planes, which are symmetric spaces associated to the exceptional simple Lie groups. As an application, we classify all Riemannian manifolds whose metric is bundle-like along the curvature constancy distribution, generalizing well-known results in Sasakian and 3-Sasakian geometry.