Cengizhan Murathan

CONTACT CR-WARPED PRODUCT SUBMANIFOLDS IN KENMOTSU SPACE FORMS

Recently, Chen studied warped products which are CR-submanifolds in Kaehler manifolds and established general sharp inequalities for CR-warped products in Kaehler manifolds. In the present paper, we obtain sharp estimates for the squared norm of the second fundamental form (an extrinsic invariant) in terms of the warping function for contact CR-warped products isometrically im- mersed in Kenmotsu space forms. The equality case is considered. Some applications are derived.

Nullity conditions in paracontact geometry

Abstract The paper is a complete study of paracontact metric manifolds for which the Reeb vector field of the underlying contact structure satisfies a nullity condition (the condition (1.2), for some real numbers κ ˜ and μ ˜ ). This class of pseudo-Riemannian manifolds, which includes para-Sasakian manifolds, was recently defined in Cappelletti Montano (2010) [13] . In this paper we show in fact that there is a kind of duality between those manifolds and contact metric ( κ , μ ) -spaces. In particular, we prove that, under some natural assumption, any such paracontact metric manifold admits a compatible contact metric ( κ , μ ) -structure (eventually Sasakian). Moreover, we prove that the nullity condition is invariant under D -homothetic deformations and determines the whole curvature tensor field completely. Finally non-trivial examples in any dimension are presented and the many differences with the contact metric case, due to the non-positive definiteness of the metric, are discussed.

Certain inequalities for submanifolds in ( K ,μ)-contact space forms

In (Ch3), B.Y. Chen established sharp relationship be- tween the Ricci curvature and the squared mean curvature for a submanifold in a Riemanian space form with arbitrary codimen- sion. In (MMO), one dealt with similar problems for submanifolds in complex space forms. In this article, we obtain sharp relationships between the Ricci curvature and the squared mean curvature for submanifolds in (k;�)-contact space forms.

BIHARMONIC SUBMANIFOLDS IN 3-DIMENSIONAL (κ,µ)-MANIFOLDS

Biharmonic maps between Riemannian manifolds are defined as critical points of the bienergy and generalized harmonic maps. In this paper, we give necessary and sufficient conditions for nonharmonic Legendre curves and anti-invariant surfaces of 3-dimensional (κ,μ)-manifolds to be biharmonic.

ON SLANT RIEMANNIAN SUBMERSIONS FOR COSYMPLECTIC MANIFOLDS

In this paper we introduce slant Riemannian submersions from cosymplectic manifolds onto Riemannian manifolds. We obtain some results on slant Riemannian submersions of a cosymplectic manifolds. We also give examples and inequalities between the scalar curvature and squared mean curvature of fibres of such slant submersions according to characteristic vector field is vertical or horizontal.

TENSOR PRODUCT SURFACES WITH POINTWISE 1-TYPE GAUSS MAP

Tensor product immersions of a given Riemannian manifold was initiated by B.-Y. Chen. In the present article we study the tensor product surfaces of two Euclidean plane curves. We show that a tensor product surface M of a plane circle c1 centered at origin with an Euclidean planar curve c2 has harmonic Gauss map if and only if M is a part of a plane. Further, we give necessary and sufficient conditions for a tensor product surface M of a plane circle c1 centered at origin with an Euclidean planar curve c2 to have pointwise 1-type Gauss map.

A study of three-dimensional paracontact (κ̃,μ̃,ν̃)-spaces

This paper is a study of three-dimensional paracontact metric (κ,μ,ν)-manifolds. Three-dimensional paracontact metric manifolds whose Reeb vector field ξ is harmonic are characterized. We focus on some curvature properties by considering the class of paracontact metric (κ,μ,ν)-manifolds under a condition which is given at Definition 3.1. We study properties of such manifolds according to the cases κ > −1, κ = −1, κ < −1 and construct new examples of such manifolds for each case. We also show the existence of paracontact metric (−1,μ≠0,ν≠0) spaces with dimension greater than 3, such that h2 = 0 but h≠0.

PSEUDO SYMMETRIC AND PSEUDO RICCI SYMMETRIC WARPED PRODUCT MANIFOLDS

We study pseudo symmetric (brie∞y (PS)n) and pseudo Ricci symmetric (brie∞y (PRS)n) warped product manifolds M £F N. If M is (PS)n, then we give a condition on the warping function that M is a pseudosymmetric space and N is a space of constant curvature. If M is (PRS)n, then we show that (i) N is Ricci symmetric and (ii) M is (PRS)n if and only if the tensor T deflned by (2.6) satisfles a certain condition.

CHEN INEQUALITIES FOR SUBMANIFOLDS OF A COSYMPLECTIC SPACE FORM WITH A SEMI-SYMMETRIC METRIC CONNECTION

Abstract In this paper, we prove Chen inequalities for submanifolds of a cosymplectic space form of constant φ-sectional curvature N2m+1(c) endowed with a semisymmetric metric connection, i.e., relations between the mean curvature associated with the semi-symmetric metric connection, scalar and sectional curvatures, k-Ricci curvature and the sectional curvature of the ambient space.

Ricci curvature of submanifolds in Kenmotsu space forms

In 1999, Chen established a sharp relationship between the Ricci curvature and the squared mean curvature for a submanifold in a Riemannian space form with arbitrary codimension. Similar problems for submanifolds in complex space forms were studied by Matsumoto et al. In this paper, we obtain sharp relationships between the Ricci curvature and the squared mean curvature for submanifolds in Kenmotsu space forms.