Generalized Complex Space Forms Research Articles

Geometric inequalities of bi-warped product submanifold in generalized complex space form

In this article, we obtain some inequalities for the bi-warped product submanifold of nearly-Kähler manifold which is expressed in terms of Laplacian, gradient, slant coefficient, and second fundamental form. Our inequality generalizes the inequalities of [1,20]. We also compute the Dirichlet energy and first eigenvalue of the Laplacian of warping functions.

Estimation of Ricci Curvature for Hemi-Slant Warped Product Submanifolds of Generalized Complex Space Forms and Their Applications

In this paper, we estimate Ricci curvature inequalities for a hemi-slant warped product submanifold immersed isometrically in a generalized complex space form with a nearly Kaehler structure, and the equality cases are also discussed. Moreover, we also gave the equivalent version of these inequalities. In a later study, we will exhibit the application of differential equations to the acquired results. In fact, we prove that the base manifold is isometric to Euclidean space under a specific condition.

Ricci curvature of semi-slant warped product submanifolds in generalized complex space forms

<abstract><p>The objective of this paper is to achieve the inequality for Ricci curvature of a semi-slant warped product submanifold isometrically immersed in a generalized complex space form admitting a nearly Kaehler structure in the expressions of the squared norm of mean curvature vector and warping function. In addition, the equality case is likewise discussed. We provide numerous physical applications of the derived inequalities. Later, we proved that under a certain condition the base manifold $ N_T^{n_1} $ is isometric to a $ n_1 $-dimensional sphere $ S^{n_1}(\frac{\lambda_1}{n_1}) $ with constant sectional curvature $ \frac{\lambda_1}{n_1}. $</p></abstract>

The eigenvalue estimates of p-Laplacian of totally real submanifolds in generalized complex space forms

The eigenvalue estimates of p-Laplacian of totally real submanifolds in generalized complex space forms

Geometric Mechanics on Warped Product Semi-Slant Submanifold of Generalized Complex Space Forms

In this study, we develop a general inequality for warped product semi-slant submanifolds of type M n = N T n 1 × f N ϑ n 2 in a nearly Kaehler manifold and generalized complex space forms using the Gauss equation instead of the Codazzi equation. There are several applications that can be developed from this. It is also described how to classify warped product semi-slant submanifolds that satisfy the equality cases of inequalities (determined using boundary conditions). Several results for connected, compact warped product semi-slant submanifolds of nearly Kaehler manifolds are obtained, and they are derived in the context of the Hamiltonian, Dirichlet energy function, gradient Ricci curvature, and nonzero eigenvalue of the Laplacian of the warping functions.

Chen-Ricci Inequalities with a Quarter Symmetric Connection in Generalized Space Forms

In this article, we obtain improved Chen-Ricci inequalities for submanifolds of generalized space forms with quarter-symmetric metric connection, with the help of which we completely characterized the Lagrangian submanifold in generalized complex space form and a Legendrian submanifold in a generalized Sasakian space form. We also discuss some geometric applications of the obtained results.

Chen–Ricci Inequality for CR-Warped Products and Related Open Problems

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.

The normalized Ricci flow and homology in Lagrangian submanifolds of generalized complex space forms

In this paper, we prove that a simply connected Lagrangian submanifold in the generalized complex space form is diffeomorphic to standard sphere [Formula: see text] and the normalized Ricci flow converges to a constant curvature metric, provided its squared norm of the second fundamental form satisfies some upper bound depending only on the squared norm of the mean curvature vector field, the constant sectional curvature, and the dimension of the Lagrangian immersion of the ambient space. Next, we conclude that stable currents do not exist and homology groups vanish in a compact real submanifold of the general complex space form, provided that the second fundamental form satisfies some extrinsic conditions. We show that our results improve some previous results.

The First Fundamental Equation and Generalized Wintgen-Type Inequalities for Submanifolds in Generalized Space Forms

In this work, we first derive a generalized Wintgen type inequality for a Lagrangian submanifold in a generalized complex space form. Further, we extend this inequality to the case of bi-slant submanifolds in generalized complex and generalized Sasakian space forms and derive some applications in various slant cases. Finally, we obtain obstructions to the existence of non-flat generalized complex space forms and non-flat generalized Sasakian space forms in terms of dimension of the vector space of solutions to the first fundamental equation on such spaces.

F-Biminimal submanifolds of generalized space forms

We study f-biminimal submanifolds in generalized complex space forms and generalized Sasakian space forms. Then, we analyze f-biminimal submanifolds in these spaces. Finally, we consider f-biminimal integral submanifolds in Sasakian space forms and give an example.

Lower extremities for generalized normalized δ-Casorati curvatures of bi-slant submanifolds in generalized complex space forms

In this paper, we obtain the inequalities for the generalized normalized δ-Casorati curvature and the normalized scalar curvature for different submanifolds in generalized complex space form, which is based on an optimization procedure involving a quadratic polynomial in the components of the second fundamental form and characterizes the submanifolds on which equalities hold. We also develop, the same inequalities for semi-slant, hemi-slant, CR, slant, invariant and anti-invariant submanifolds in the same target space and consider the equality case. Moreover, we obtain a geometric inequality involving Casorati curvature for warped product bi-slant submanifolds in same ambient and obtain an obstruction result.

CR-warped product submanifolds of a generalized complex space form

In this paper CR-warped product submanifolds of a generalized complex space form are studied and a characterizing inequality for existence of CR-warped product submanifolds is established. Moreover, some special cases are also discussed.

On generalized complex space forms satisfying certain curvature conditions

We study Ricci soliton $(g,V,\lambda)$ of generalized complex space forms when the Riemannian, Bochner and $W_{2}$ curvature tensors satisfy certain curvature conditions like semi-symmetric, Einstein semi-symmetric, Ricci pseudo symmetric and Ricci generalized pseudo symmetric. In this study it is shown that shrinking, steady and expansion of the generalized complex space forms depends on the solenoidal property of vector $V$. Also we prove that generalized complex space form with conservative Bochner curvature tensor is constant scalar curvature.

Biharmonic submanifolds of generalized space forms

We consider biharmonic submanifolds in both generalized complex and Sasakian space forms. After giving the biharmonicity conditions for submanifolds in these spaces, we study different particular cases for which we obtain curvature estimates. We consider curves, complex and Lagrangian surfaces and hypersurfaces for the generalized complex space form as well as hypersurfaces, invariant and anti-invariant submanifolds in the case of generalized Sasakian space forms.

Optimal Inequalities for the Casorati Curvatures of Submanifolds in Generalized Space Forms Endowed with Semi-Symmetric Non-Metric Connections

In this paper, we prove some optimal inequalities involving the intrinsic scalar curvature and the extrinsic Casorati curvature of submanifolds in a generalized complex space form with a semi-symmetric non-metric connection and a generalized Sasakian space form with a semi-symmetric non-metric connection. Moreover, we show that in both cases, the equalities hold if and only if submanifolds are invariantly quasi-umbilical.

Totally real submanifolds in a generalized complex space form

In the present paper, we investigate totally real submanifolds in generalized complex space form. We study the [Formula: see text]-structure in the normal bundle of a totally real submanifold and derive some integral formulas computing the Laplacian of the square of the second fundamental form and using these formulas, we prove a pinching theorem. In fact, the purpose of this note is to generalize results proved in B. Y. Chen and K. Ogiue, On totally real manifolds, Trans. Amer. Math. Soc. 193 (1974) 257–266, S. S. Chern, M. Do Carmo and S. Kobayashi, Minimal submanifolds of a sphere with second fundamental form of constant length, in Functional Analysis and Related Fields (Springer-Verlag, 1970), pp. 57–75 to the case, when the ambient manifold is generalized complex space form.

A useful orthonormal basis on bi-slant submanifolds of almost Hermitian manifolds

In this paper, we study bi-slant submanifolds of an almost Hermitian manifold for different cases. We introduce a new orthonormal basis on bi-slant submanifold, semi-slant submanifold and hemi-slant submanifold of an almost Hermitian manifold to compute Chen's main inequalities. We investigate these inequalities for semi-slant submanifolds, hemi-slant submanifolds and slant submanifolds of a generalized complex space form. We obtain some characterizations on such submanifolds of a complex space form.

Generalized Sasakian Space Forms and Riemannian Manifolds of Quasi Constant Sectional Curvature

In this paper, we show that a generalized Sasakian space form of dimension greater than three is either of constant sectional curvature; or a canal hypersurface in Euclidean or Minkowski spaces; or locally a certain type of twisted product of a real line and a flat almost Hermitian manifold; or locally a wapred product of a real line and a generalized complex space form; or an $\alpha$-Sasakian space form; or it is of five dimension and admits an $\alpha$-Sasakian Einstein structure. In particular, a local classification for generalized Sasakian space forms of dimension greater than five is obtained. A local classification of Riemannian manifolds of quasi constant sectional curvature of dimension greater than three is also given in this paper.

Chen Inequalities For Submanifolds Of Generalized Space Forms With a Semi-symmetric Metric Connection

We investigate sharp inequalities for submanifolds in both generalized complex space forms and generalized Sasakian space forms with a semisymmetric metric connection.

A study on hypersurface of complex space form

We show that quasi-umbilical, generalized quasi-umbilical, super quasi-umbilical hypersurfaces of a complex space form are quasi-Einstein, mixed generalized quasi-Einstein and mixed super quasi-Einstein manifolds, respectively. We also prove that a Bochner at space of generalized complex space form is an Einstein manifold.