In this paper, we study warped product pseudo-slant submanifolds of nearly Kaehler manifolds. We prove the non-existence results on warped product submanifolds of a nearly Kaehler manifold.

The fundamental theorem of submanifolds is adapted to space-times. It is shown that the integrability conditions for the existence of submanifolds of a pseudo-Euclidean space contain the Einstein and Yang-Mills equations.

We show the volume maximizing property of the special Lagrangian submanifolds of a pseudo-Euclidean space. These special Lagrangian submanifolds arise locally as gradient graphs of solutions to MongeAmpère equations.

It was conjectured in 1 II] (also in [2]) that mixed foliate CR-submanifolds in a complex hyperbolic space are either complex submanffolds or totally real submanifolds. In this paper we give an affirmative solution to this conjecture.

We prove the existence of contact submanifolds realizing the Poincaré dual of the top Chern class of a complex vector bundle over a closed contact manifold. This result is analogue in the contact category to Donaldson’s construction of symplectic submanifolds. The main tool in the construction is to show the existence of sequences of sections which are asymptotically holomorphic in an appropiat...

We introduce the notion of twisted generalized complex submanifolds and describe an equivalent characterization in terms of Poisson-Dirac submanifolds. Our characterization recovers a result of Vaisman [21]. An equivalent characterization is also given in terms of spinors. As a consequence, we show that the fixed locus of an involution preserving a twisted generalized complex structure is a twi...

In the present paper we survey the most recent classification results for proper biharmonic submanifolds in unit Euclidean spheres. We also obtain some new results concerning geometric properties of proper biharmonic constant mean curvature submanifolds in spheres.