On submersions of the complex indicatrix

In this paper we study the complex indicatrix associated to a complex Finsler space as an embedded CR - hypersurface of the holomorphic tangent bundle, considered in a fixed point. Following the study of CR - submanifolds of a K?hler manifold, there are investigated some properties of the complex indicatrix as a real submanifold of codimension one, using the submanifold formulae and the fundamental equations. As a result, the complex indicatrix is an extrinsic sphere of the holomorphic tangent space in each fibre of a complex Finsler bundle. Also, submersions from the complex indicatrix onto an almost Hermitian manifold and some properties that can occur on them are studied. As application, an explicit submersion onto the complex projective space is provided.

On lightlike geometry in indefinite Kenmotsu manifolds

We investigate some geometric aspects of lightlike hypersurfaces of indefinite Kenmotsu manifolds, tangent to the structure vector field, by paying attention to the geometry of leaves of integrable distributions. Theorems on parallel vector fields, Killing distribution, geodesibility of their leaves are obtained. The geometric configuration of such lightlike hypersurfaces and leaves of its screen integrable distributions are established. We show that no totally contact umbilical leaf of a screen integrable distribution of a lightlike hypersurface can be an extrinsic sphere. We also prove that the geometry of any leaf of an integrable distribution is closely related to the geometry of a normal bundle.

On totally umbilical QR-submanifolds of quaternion Kaehlerian manifolds

We introduce the notion of generalised 3-Sasakian structure on a manifold and show that a totally umbilical, but not totally geodesic, proper QR-submanifold of a quaternion Kaehlerian manifold is an extrinsic sphere and inherits such a structure.

CR-hypersurfaces of the six-dimensional sphere

We proved that there does not exist a properCR-hypersurface ofS6with parallel second fundamental form. As a result of this we showed thatS6does not admit a properCR-totally umbilical hypersurface. We also proved that an Einstein properCR-hypersurface ofS6is an extrinsic sphere.