Clifford structures on Riemannian manifolds

Details Clifford structures on Riemannian manifolds Clifford structures on Riemannian manifolds

2009-12-21

arxiv.org/abs/0912.4207v4

Adv. Math. 228, 940-967 (2011)

Mathematics

Differential Geometry

Final version, 28 pages

Scientific paper

10.1016/j.aim.2011.06.006

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.

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