Almost Complex Surfaces in the Nearly Kähler SL(2,ℝ) × SL(2,ℝ)

The space S L ( 2 , R ) × S L ( 2 , R ) admits a natural homogeneous pseudo-Riemannian nearly Kähler structure. We investigate almost complex surfaces in this space. In particular, we obtain a complete classification of the totally geodesic almost complex surfaces and of the almost complex surfaces with parallel second fundamental form.

Some results on Finsler submanifolds

In this paper we study the submanifold theory in terms of Chern connection. We introduce the notions of the second fundamental form and mean curvature for Finsler submanifolds, and establish the fundamental equations by means of moving frame for the hypersurface case. We provide the estimation of image radius for compact submanifold, and prove that there exists no compact minimal submanifold in any complete noncompact and simply connected Finsler manifold with nonpositive flag curvature. We also characterize the Minkowski hyperplanes, Minkowski hyperspheres and Minkowski cylinders as the only hypersurfaces in Minkowski space with parallel second fundamental form.

CR-submanifolds with the symmetric σ in a locally conformal Kaehler space form

In this paper, we consider CR-submanifolds with the symmetric ?? which is a generalization of parallel second fundamental form, in a locally conformal Kaehler space form. About the symmetric tensor field P defined in (1.7), we show that, in an anti-holomorphic submanifold in an l.c.K.-space form, P is diagonal with respect to an adapted frame and has two eigenfunctions (See Theorem 3.1). Finally, we consider the relation of the eigenfunctions of P and the Lee form (See Theorems 3.2 and 3.3).