We describe how the loop group maps corresponding to special submanifolds associated to integrable systems may be thought of as certain Grassmann submanifolds of infinite dimensional homogeneous spaces. In general, the associated families of special submanifolds are certain Grassmann submanifolds. An example is given from the recent article [2].

We study families of submanifolds in symmetric spaces of compact type arising as exponential images of s-orbits of variable radii. If the s-orbit is symmetric such submanifolds are the most important examples of adapted submanifolds, i.e. of submanifolds of symmetric spaces with curvature invariant tangent and normal spaces.

The main goal of our paper is the study of several classes of submanifolds of generalized complex manifolds. Along with the generalized complex submanifolds defined by Gualtieri and Hitchin in [4], [8] (we call these “generalized Lagrangian submanifolds” in our paper), we introduce and study three other classes of submanifolds. For generalized complex manifolds that arise from complex (resp., s...

The class of generic submanifold includes all real hypersurfaces, complex submanifolds, totally real submanifolds, and CR-submanifolds. In this paper we initiate the study of generic submanifolds in a nearly Kaehler manifold from differential geometric point of view. Some fundamental results in this paper will be obtained.

In this paper, we study geodesic contact CR-lightlike submanifolds and geodesic screen CR-lightlike (SCR) submanifolds of indefinite Sasakian manifolds. Some necessary and sufficient conditions for totally geodesic, mixed geodesic, D -geodesic and -geodesic contact CR-lightlike submanifolds and SCR submanifolds are obtained. D

The geometry of CR-submanifolds of Kaehler manifolds was initiated by Bejancu 1 and has been developed by 2–5 and others. They studied the geometry of CR-submanifolds with positive definite metric. Thus, this geometry may not be applicable to the other branches of mathematics and physics, where the metric is not necessarily definite. Moreover, because of growing importance of lightlike submanif...

The main goal of our paper is the study of several classes of submanifolds of generalized complex manifolds. Along with the generalized complex submanifolds defined by Gualtieri and Hitchin in [Gua], [H3] (we call these “generalized Lagrangian submanifolds” in our paper), we introduce and study three other classes of submanifolds and their relationships. For generalized complex manifolds that a...

We first prove some results on invariant lightlike submanifolds of indefinite Sasakian manifolds. Then, we introduce a general notion of contact Cauchy-Riemann (CR) lightlike submanifolds and study the geometry of leaves of their distributions. We also study a class, namely, contact screen Cauchy-Riemann (SCR) lightlike submanifolds which include invariant and screen real subcases. Finally, we ...

Having fixed a Kaehler class and the unique corresponding hyperkaehler metric, we prove that all special Lagrangian submanifolds of an irreducible symplectic 4-fold X are obtained by complex submanifolds via a generalization of the so called hyperkaehler rotation trick; thus they retain part of the rigidity of the complex submanifolds: indeed all special Lagrangian submanifolds of X turn out to...

Given a real vector space V equipped with an Euclidean metric, (after rescaling) any p-form φ ∈ V p V ∗ defines a calibration on V . This note identifies an exterior differential system whose integral submanifolds are precisely the critical submanifolds of the calibration. In particular, calibrated submanifolds are necessarily integral submanifolds of the system. The result is extended to calib...