Extensions of CR-Structures to Other Geometrical Structures

Abstract

Let N be a real (2n + l)-dimensional almost contact metric manifold with structure tensors (φ, ξ, n, g), where φ is a tensor field of type (1, 1), ξ is a vector field, n, is a 1-form and g is a Riemannian metric on N. These tensor fields are related by (see §of Chapter I) $${{\phi }^{2}}X = - X + \eta (X)\xi ; \phi \xi = 0; \eta (\xi ) = 1; \eta (\phi X) = 0$$ (1.1) $$g(\phi X,\phi Y) = g(X,Y) - \eta (X) \cdot \eta (Y); \eta (X) = g(X,\xi ),$$ (1.2) for any vector fields X, Y tangent to N.