Clifford structures on Riemannian manifolds |
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Authors: | Andrei Moroianu Uwe Semmelmann |
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Institution: | aCMLS, École Polytechnique, UMR 7640 du CNRS, 91128 Palaiseau, France;bInstitut für Geometrie und Topologie, Fachbereich Mathematik, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany |
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Abstract: | 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 OP2, (C⊗O)P2, (H⊗O)P2 and (O⊗O)P2, 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. |
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Keywords: | MSC: 53C26 53C35 53C10 53C15 |
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