Abstract
N. Tanaka ([10]) defined the canonical affine connection on a nondegenerate integrable CR manifold. In the present paper, we introduce a new class of contact Riemannian manifolds satisfying (C) (\((C)(\hat \nabla \dot \gamma R)( \cdot ,\dot \gamma )\dot \gamma = 0\) for any unit\(\hat \nabla \)-geodesic (\(\gamma (\hat \nabla _{\dot \gamma } \dot \gamma = 0)\), where\(\hat \nabla \) is the generalized Tanaka connection. In particular, when the associated CR structure of a given contact Riemannian manifold is integrable we have a structure theorem and find examples which are neither Sasakian nor locally symmetric but satisfy the condition (C).
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This work was supported in part by BSRI 98-1425.
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Cho, J.T. A new class of contact riemannian manifolds. Isr. J. Math. 109, 299–318 (1999). https://doi.org/10.1007/BF02775040
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DOI: https://doi.org/10.1007/BF02775040