Warped Product CR-Submanifolds of Lorentzian -Kenmotsu Manifolds
Warped product semi-invariant submanifolds in almost paracontact Riemannian manifolds.
Warped product submanifolds of cosymplectic manifolds.
Willmore submanifolds in the unit sphere.
In this paper we generalize the self-adjoint differential operator (used by Cheng-Yau) on hypersurfaces of a constant curvature manifold to general submanifolds. The generalized operator is no longer self-adjoint. However we present its adjoint operator. By using this operator we get the pinching theorem on Willmore submanifolds which is analogous to the pinching theorem on minimal submanifold of a sphere given by Simon and Chern-Do Carmo-Kobayashi.
Witt algebra and the curvature of the Heisenberg group
The aim of this paper is to determine explicitly the algebraic structure of the curvature algebra of the 3-dimensional Heisenberg group with left invariant cubic metric. We show, that this curvature algebra is an infinite dimensional graded Lie subalgebra of the generalized Witt algebra of homogeneous vector fields generated by three elements.