Clifford Structures on Riemannian Manifolds

We introduce the notion of even Clifford structures on Riemannian manifolds, which for rank r = 2 and r = 3 reduce to almost Hermitian and quaternion-Hermitian structures respectively. We give the complete classification of manifolds carrying parallel rank r even Clifford structures: Kähler, quaternion-Kähler and Riemannian products of quaternion-Kähler manifolds for r = 2, 3 and 4 respectively, several classes of 8-dimensional manifolds (for 5 ≤ r ≤ 8), families of real, complex and quaternionic Grassmannians (for r = 8, 6 and 5 respectively), and Rosenfeld’s elliptic projective planes OP, (C ⊗ O)P, (H⊗O)P and (O⊗O)P, which are symmetric spaces associated to the exceptional simple Lie groups F4, E6, E7 and E8 (for r = 9, 10, 12 and 16 respectively). 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. 2000 Mathematics Subject Classification: Primary 53C26, 53C35, 53C10, 53C15.