Luis M. Fernández

SLANT SUBMANIFOLDS IN SASAKIAN MANIFOLDS

Abstract. In this paper, we show new results on slant submanifolds of analmost contact metric manifold. We study and characterize slant submanifolds of K-contact and Sasakian manifolds. We also study the special class of three-dimen-sional slant submanifolds. We give several examples of slant submanifolds.1991 Mathematics Subject Classification. 53C15, 53C40.0. Introduction. Slant immersions in complex geometry were defined by B.-Y.Chen as a natural generalization of both holomorphic immersions and totally realimmersions [2]. Examples of slant immersions into complex Euclidean spaces C 2 andC 4 were given by Chen and Tazawa [2, 4, 5], while slant immersions of Ka¨hler C-spaces into complex projective spaces were given by Maeda, Ohnita and Udagawa[9].In a recent paper [7], A. Lotta has introduced the notion of slant immersion of aRiemannian manifold into an almost contact metric manifold and he has provedsome properties of such immersions. A. Lotta and A. M. Pastore have obtainedexamples of slant submanifolds in the Sasakian-space-form R

Semi-Slant Submanifolds of a Sasakian Manifold

We define and study both bi-slant and semi-slant submanifolds of an almost contact metric manifold and, in particular, of a Sasakian manifold. We prove a characterization theorem for semi-slant submanifolds and we obtain integrability conditions for the distributions which are involved in the definition of such submanifolds. We also study an interesting particular class of semi-slant submanifolds.

The Structure of a Class of K-contact Manifolds

In this paper we study a class of K-contact manifolds, namely φ-conformally flat K-contact manifolds and we show that a compact φ-conformally flat K-contact manifold with regular contact vector field is a principal S1-bundle over an almost Kaehler space of constant holomorphic sectional curvature.

Minimal slant submanifolds of the smallest dimension in S-manifolds

We study slant submanifolds of S-manifolds with the smallest dimension, specially minimal submanifolds and establish some relations between them and anti-invariant submanifolds in S-manifolds, similar to those ones proved by B.-Y. Chen for slant surfaces and totally real surfaces in Kaehler manifolds.

MORSE INEQUALITIES ON CERTAIN INFINITE 2-COMPLEXES

Using the notion of discrete Morse function introduced by R. Forman for finite cw -complexes, we generalize it to the infinite 2-dimensional case in order to get the corresponding version of the well-known discrete Morse inequalities on a non-compact triangulated 2-manifold without boundary and with finite homology. We also extend them for the more general case of a non-compact triangulated 2-pseudo-manifold with a finite number of critical simplices and finite homology.

The curvature of submanifolds of anS-space form

Many authors have studied the geometry of submanifolds of Kaehlerian and Sasakian manifolds. On the other hand, David E. Blair has initiated the study of S-manifolds, which reduce, in particular cases, to Sasakian manifolds [1]. I. Mihai [7] and Ornea [8] have studied CR-submanifolds of S-manifolds. The purpose of the present paper is to investigate some properties ofinvariant and anti-invariant submanifolds of an S-manifold whose invariant f-sectional curvature is constant, that is, of an S-space form. Specifically, those ones related with the curvature tensor fields and with the scalar curvature on the submanifold. In Section 1 we review basic formulas for submanifolds in Riemannian manifolds and, in Section 2, for S-manifolds. In Sections 3 and 4 we study anti-invariant and invariant submanifolds, respectively, of an S-space form. Finally, in the last section we give some examples.

New relationships involving the mean curvature of slant submanifolds in S-space-forms

Relationships between the Ricci curvature and the squared mean curvature and between the shape operator associated with the mean curvature vector and the sectional curvature function for slant submanifolds of an S-space-form are proved, particularizing them to invariant and anti-invariant submanifolds tangent to the structure vector fields.

The structure of some classes of contact metric manifolds

Abstract We study the conformal curvature tensor and the contact conformal curvature tensor in Sasakian and/or K-contact manifolds. We find a necessary and sufficient condition for a Sasakian manifold to be φ-conformally flat. We also find some necessary conditions for a K-contact manifold to be φ-contact conformally flat. Then we give a structure theorem for φ-contact conformally flat Sasakian manifolds. It is also proved that a Sasakian manifold cannot be ξ-contact conformally flat.

Submanifolds associated with graphs

In this paper we present an interesting relationship between graph theory and differential geometry by defining submanifolds of almost Hermitian manifolds associated with certain kinds of graphs. We show some results about the possibility of a graph being associated with a submanifold and we use them to characterize CR-submanifolds by means of trees. Finally, we characterize submanifolds associated with graphs in a four-dimensional almost Hermitian manifold.

The ℳ-projective curvature tensor field on generalized (κ,μ)-paracontact metric manifolds

Abstract We consider generalized ( κ , μ ) {(\kappa,\mu)} -paracontact metric manifolds satisfying certain flatness conditions on the ℳ {\mathcal{M}} -projective curvature tensor. Specifically, we study ξ- ℳ {\mathcal{M}} -projectively flat and ℳ {\mathcal{M}} -projectively flat generalized ( κ , μ ) {(\kappa,\mu)} -paracontact metric manifolds and, further, ϕ- ℳ {\mathcal{M}} -projectively symmetric generalized ( κ ≠ - 1 , μ ) {(\kappa\neq-1,\mu)} -paracontact metric manifolds. We prove that they are characterized by certain structures whose properties are discussed in some detail.