Ion Mihai

Contact CR-Warped Product Submanifolds in Sasakian Manifolds

Recently, B.-Y. 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 a sharp inequality for the squared norm of the second fundamental form (an extrinsic invariant) in terms of the warping function for contact CR-warped products isometrically immersed in Sasakian manifolds. The equality case is considered. Also, the minimum codimension of a contact CR-warped product in an odd-dimensional sphere is determined.

RICCI CURVATURE OF SUBMANIFOLDS IN SASAKIAN SPACE FORMS

Recently, 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. Afterwards, we dealt with similar problems for submanifolds in complex space forms. In the present paper, we obtain sharp inequalities between the Ricci curvature and the squared mean curvature for submanifolds in Sasakian space forms. Also, estimates of the scalar curvature and the k-Ricci curvature respectively, in terms of the squared mean curvature, are proved.

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.

Tensor product surfaces of euclidean plane curves

Recently B.Y. CHEN initiated the study of the tensor product immersion of two immersions of a given Riemannian manifold [3]. In [6] the particular case of tensor product of two Euclidean plane curves was studied. The minimal one were classified, and necessary and sufficient conditions for such a tensor product to be totally real or complex or slant were established. In the present paper we study for tensor product of Euclidean plane curves the problem of B.Y. CHEN: to what extent do the properties of the tensor product immersion f ⊗ h of two immersions f, h determines the immersions f, h ? [3]

On 3-dimensional contact slant submanifolds in Sasakian space forms

Recently, B.-Y. Chen obtained an inequality for slant surfaces in complex space forms. Further, B.-Y. Chen and one of the present authors proved the non-minimality of proper slant surfaces in non-flat complex space forms. In the present paper, we investigate 3-dimensional proper contact slant submanifolds in Sasakian space forms. A sharp inequality is obtained between the scalar curvature (intrinsic invariant) and the main extrinsic invariant, namely the squared mean curvature. It is also shown that a 3-dimensional contact slant submanifold M of a Sasakian space form ( c ), with c ≠ 1, cannot be minimal.

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.

A class of C-totally real submanifolds of Sasakian space forms

Recently, Chen defined an invariant &M of a Riemannian manifold M. Sharp inequalities for this Riemannian invariant were obtained for submanifolds in real, complex and Sasakian space forms, in terms of their mean curvature. In the present paper, we investigate certain C-totally real submanifolds of a Sasakian space form M 2m+ \c) satisfying Chens equality.

Chapter 6 Complex differential geometry

Publisher Summary The geometry of complex manifolds, in particular Kaehler manifolds, is an important research topic in Differential Geometry. The chapter presents the basic notions and certain important results in complex differential geometry. It defines complex and almost complex manifolds and gives standard examples. Locally, a Kaehlerian metric differs from the Euclidean metric on the complex space C n starting with the second power of the Taylor series. There are topological obstructions to the existence of Kaehlerian metrics on a compact complex manifold. The differential forms on a Kaehler manifold are studied and the complex version of Hodge theorem is proved in the chapter.

On space-time carrying a total hyperbolic skew symmetric Killing vector field

The complex vectorial formalism on a general space–time (M,g) was constructed by Cahen, Debever and Defrise. This formalism is based on the local isomorphism I:L(4)→SO3(C), where L(4) is the four-dimensional Lorentz group acting on the tangent spaces TpM and SO3(C) is the three-dimensional complex rotation group. In this framework, the congruence of Debever plays a distinguished role. Its properties determine the general space–time M, in terms of Petrov’s classification. In the present paper, we assume that any hyperbolic vector field X on M is a skew symmetric Killing vector field having a spatial vector field Y as generative. The existence of such a vector field X is determined by an exterior differential system in involution. It is shown that M is the local Riemannian product M=Mh×Ms, where Mh (resp. Ms) is a totally geodesic and totally pseudo-isotropic hyperbolic (resp. spatial) surface (the Gauss map is ametric). Any such M is a space–time of type D in Petrov’s classification. It is proved that the congruence of Debever is of electric type; in particular, it is geodesic and shear 1-free. Other geometric properties on such a general space–time are obtained.

An improved first Chen inequality for Legendrian submanifolds in Sasakian space forms

Legendrian submanifolds in Sasakian space forms play an important role in contact geometry. Defever et al. (Boll Unione Mat Ital B 7(11):365–374, 1997) established the first Chen inequality for C-totally real submanifolds in Sasakian space forms. In this article, we improve this first Chen inequality for Legendrian submanifolds in Sasakian space forms.