CR-warped product submanifolds of a generalized complex space form

As warped product manifolds provide an excellent setting to model space time near black holes or bodies with large gravitational field, the study of these manifolds assumes significance in general. B. Y. Chen [4] initiated the study of CR-warped product submanifolds in a Kaehler manifold. He obtained a characterization for a CR-submanifold to be locally a CR-warped product and an estimate for the squared norm of the second fundamental form of CR-warped products in a complex space form (cf [6]). In the present paper, we have obtained a necessary and sufficient conditions in terms of the canonical structures P and F on a CR-submanifold of a nearly Kaehler manifold under which the submanifold reduces to a locally CR-warped product submanifold. Moreover, an estimate for the second fundamental form of the submanifold in a generalized complex space is obtained and thus extend the results of Chen to a more general setting.

In this paper, we study CR-warped product submanifolds of [Formula: see text]-manifolds. We prove that the CR-warped product submanifolds with invariant fiber are trivial warped products and provide a characterization theorem of CR-warped products with anti-invariant fiber of [Formula: see text]-manifolds. Moreover, we develop an inequality of CR-warped product submanifolds for the second fundamental form in terms of warping function and the equality cases are considered. Also, we find a necessary and sufficient condition for compact oriented CR-warped products turning into CR-products of [Formula: see text]-space forms.

The fundamental goal of this study was to achieve the Ricci curvature inequalities for a skew CR-warped product (SCR W-P) submanifold isometrically immersed in a complex space form (CSF) in the expressions of the squared norm of mean curvature vector and warping functions (W-F). The equality cases were likewise examined. In particular, we also derived Ricci curvature inequalities for CR-warped product (CR W-P) submanifolds. To sustain this study, an example of these submanifolds is provided.

We obtain a necessary condition for homology group to be zero on CR-warped product submanifold in Euclidean spaces in terms of second fundamental form of the submanifold and warping function. By using this condition, we show that such CR-warped product submanifold is a homotopy sphere.

The warped product \(N_1\times _f N_2\) of two Riemannian manifolds \((N_1,g_1)\) and \((N_2,g_2)\) is the product manifold \(N_1\times N_2\) equipped with the warped product metric \(g=g_1+f^2 g_2\), where f is a positive function on \(N_1\). Warped products play very important roles in differential geometry as well as in physics. A submanifold M of a Kaehler manifold \(\tilde{M}\) is called a CR-warped product if it is a warped product \(M_T\times _f N_\perp \) of a complex submanifold \(M_T\) and a totally real submanifold \(M_\perp \) of \(\tilde{M}\). In this article we survey recent results on warped product and CR-warped product submanifolds in Kaehler manifolds. Several closely related results will also be presented.

Generic warped product submanifolds of locally conformal keahler manifolds

The main objective of this paper is to achieve the Chen-Ricci inequality for skew CR-warped product submanifold isometrically immersed in a Cosymplectic space form in the expressions of the squared norm of mean curvature vector and warping functions. The equality cases is likewise discussed. However, in particular, we also derive Chen-Ricci inequality for CR-warped product submanifolds.

In this paper, taking into account that the 6-dimensional unit sphere is nearly Kaehler manifold, 4-dimensional CR-warped product submanifolds of the sphere are studied. First, an interesting relation is obtained among the warping function of the CR-warped product submanifold, the scalar curvature of the fibers and the components of the second fundamental form. Using this relation, topological and differential sphere theorems are given and totally geodesicity of CR-warped product submanifold of the 6-dimensional sphere is obtained. Moreover, a result is presented about homology groups of a CR-warped submanifold.

Warped product manifolds are known to have applications in Physics. For instance, they provide an excellent setting to model space-time near a black hole or a massive star (cf. [HONG, S. T.: Warped products and black holes, Nuovo Cimento Soc. Ital. Fis. B 120 (2005), 1227–1234]). The studies on warped product manifolds with extrinsic geometric point of view are intensified after B. Y. Chen’s work on CR-warped product submanifolds of Kaehler manifolds. Later on, similar studies are carried out in the setting of Sasakian manifolds by Hasegawa and Mihai. As Kenmotsu manifolds are themselves warped product spaces, it is interesting to investigate warped product submanifolds of Kenmotsu manifolds. In the present note a larger class of warped product submanifolds than the class of contact CR-warped product submanifolds is considered. More precisely the existence of warped product submanifolds of a Kenmotsu manifold with one of the factors an invariant submanifold is ensured, an example of such submanifolds is provided and a characterization for a contact CR-submanifold to be a contact CR-warped product submanifold is established.

In this article, we derived an equality for CR-warped product in a complex space form which forms the relationship between the gradient and Laplacian of the warping function and second fundamental form. We derived the necessary conditions of a CR-warped product submanifolds in Ka¨hler manifold to be an Einstein manifold in the impact of gradient Ricci soliton. Some classification of CR-warped product submanifolds in the Ka¨hler manifold by using the Euler–Lagrange equation, Dirichlet energy and Hamiltonian is given. We also derive some characterizations of Einstein warped product manifolds under the impact of Ricci Curvature and Divergence of Hessian tensor.

We survey recent results on CR-submanifolds in complex space forms and contact CR-submanifolds in Sasakian space forms, including a few contributions of the present authors. The Ricci curvature and k-Ricci curvature of such submanifolds are estimated in terms of the squared mean curvature. A Wintgen-type inequality for totally real surfaces in complex space forms is proved. The equality case is shown to hold if and only if the ellipse of curvature is a circle at every point of the surface; an example of a totally real surface in \(\mathbf{C}^2\) satisfying the equality case identically is provided. A generalized Wintgen inequality for Lagrangian submanifolds in complex space forms is established. A Wintgen-type inequality for CR-submanifolds in complex space forms is stated. Geometric inequalities for warped product submanifolds in complex space forms are proved, the equality cases are characterized and examples for the equality cases are given. Also some obstructions to the minimality of warped product CR-submanifolds in complex space forms are derived. The scalar curvature of such submanifolds is estimated and classifications of submanifolds in complex space forms satisfying the equality case are given. The survey ends with results on contact CR-submanifolds in Sasakian space forms. Geometric inequalities for the Ricci curvature and k-Ricci curvature of contact CR-submanifolds in Sasakian space forms are stated. A recent result by the second author (i.e., a generalized Wintgen inequality for C-totally real submanifolds in Sasakian space forms) is extended to contact CR-submanifolds in Sasakian space forms.

Chen–Ricci inequality is derived for CR-warped products in complex space forms, Theorem 4.1, involving an intrinsic invariant (Ricci curvature) controlled by extrinsic one (the mean curvature vector), which provides an answer for Problem 1. As a geometric application, this inequality is applied to derive a necessary condition for the immersed submanifold to be minimal in a complex Euclidean space, which presents a partial answer for the well-known problem proposed by S.S. Chern, Problem 2. Moreover, various applications are given. In addition, a rich geometry of CR-warped products appeared when the equality cases are discussed. Also, we extend this inequality to generalized complex space forms. In further research directions, we address a couple of open problems, namely Problems 3 and 4.

In the present paper, we determine the holomorphic curvature tensor of generalized complex space forms and study some properties of this tensor in generalized complex space forms. Moreover, we present results on generalized complex space forms satisfying curvature identities named Walker type identities.In the present paper, we determine the holomorphic curvature tensor of generalized complex space forms and study some properties of this tensor in generalized complex space forms. Moreover, we present results on generalized complex space forms satisfying curvature identities named Walker type identities.

In this paper, we obtain an inequality about Ricci curvature and squared mean curvature of slant submanifolds in generalized complex space forms. We also obtain an inequality about the squared mean curvature and the normalized scalar curvature of slant submanifolds in generalized coplex space forms.

<abstract> The main objective of this paper is to achieve the Chen-Ricci inequality for biwarped product submanifolds isometrically immersed in a complex space form in the expressions of the squared norm of mean curvature vector and warping functions.The equality cases are likewise discussed. In particular, we also derive Chen-Ricci inequality for CR-warped product submanifolds and point wise semi slant warped product submanifolds. </abstract>