Submanifolds In Complex Space Forms Research Articles

Rigidity results on totally real submanifolds in complex space forms

Rigidity results on totally real submanifolds in complex space forms

The eigenvalues of $ \beta $-Laplacian of slant submanifolds in complex space forms

<abstract><p>In this paper, we provided various estimates of the first nonzero eigenvalue of the $ \beta $-Laplacian operator on closed orientated $ p $-dimensional slant submanifolds of a $ 2m $-dimensional complex space form $ \widetilde{\mathbb{V}}^{2m}(4\epsilon) $ with constant holomorphic sectional curvature $ 4\epsilon $. As applications of our results, we generalized the Reilly-inequality for the Laplacian to the $ \beta $-Laplacian on slant submanifolds of a complex Euclidean space and a complex projective space.</p></abstract>

P-harmonic 1-forms on totally real submanifolds in complex space forms

In this paper, we consider an n-dimensional totally real submanifold M in complex space forms. We establish some vanishing type theorems for p-harmonic 1-forms on such submanifolds in both cases: either M is minimal; or M is non-minimal.

Differential Geometry of Submanifolds in Complex Space Forms Involving δ-Invariants

One of the fundamental problems in the theory of submanifolds is to establish optimal relationships between intrinsic and extrinsic invariants for submanifolds. In order to establish such relations, the first author introduced in the 1990s the notion of δ-invariants for Riemannian manifolds, which are different in nature from the classical curvature invariants. The earlier results on δ-invariants and their applications have been summarized in the first author’s book published in 2011 Pseudo-Riemannian Geometry, δ-Invariants and Applications (ISBN: 978-981-4329-63-7). In this survey, we present a comprehensive account of the development of the differential geometry of submanifolds in complex space forms involving the δ-invariants done mostly after the publication of the book.

The first eigenvalue for the p-Laplacian on Lagrangian submanifolds in complex space forms

The goal of this paper is to prove new upper bounds for the first positive eigenvalue of the [Formula: see text]-Laplacian operator in terms of the mean curvature and constant sectional curvature on Riemannian manifolds. In particular, we provide various estimates of the first eigenvalue of the [Formula: see text]-Laplacian operator on closed orientate [Formula: see text]-dimensional Lagrangian submanifolds in a complex space form [Formula: see text] with constant holomorphic sectional curvature [Formula: see text]. As applications of our main theorem, we generalize the Reilly-inequality for the Laplacian [R. C. Reilly, On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space, Comment. Math. Helv. 52(4) (1977) 525–533] to the [Formula: see text]-Laplacian for a Lagrangian submanifold in a complex Euclidean space and complex projective space for [Formula: see text] and [Formula: see text], respectively.

Conformally flat, minimal, Lagrangian submanifolds in complex space forms

Conformally flat, minimal, Lagrangian submanifolds in complex space forms

Gap theorems for Lagrangian submanifolds in complex space forms

Gap theorems for Lagrangian submanifolds in complex space forms

Optimal inequalities for contact CR-submanifolds in almost contact metric manifolds

Abstract In [An optimal inequality for CR-warped products in complex space forms involving CR δ-invariant, Internat. J. Math. 23(3) (2012)], B.-Y. Chen introduced the CR δ-invariant for CR-submanifolds. Then, in [Two optimal inequalities for anti-holomorphic submanifolds and their applications, Taiwan. J. Math. 18 (2014), 199–217], F. R. Al-Solamy, B.-Y. Chen and S. Deshmukh proved two optimal inequalities for anti-holomorphic submanifolds in complex space forms involving the CR δ-invariant. In this paper, we obtain optimal inequalities for this invariant for contact CR-submanifolds in almost contact metric manifolds.

Chen-Ricci inequality for biwarped product submanifolds in complex 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>

An integral formula for bi-slant submanifolds in complex space forms

The present paper deals with bi-slant submanifolds in complex space forms. We obtain an integral formula of Simons' type for bi-slant submanifolds of a complex space form with positive constant holomorphic sectional curvature. Then, we apply it to prove our main result. We also discuss the same result in different submanifolds such as hemi-slant and semi-slant submanifolds.

Certain submanifolds of complex space forms

After recalling the geometric meaning of the commutativity of the second fundamental tensor of a real hypersurface of a complex space form and its induced almost contact structure, we present some classification theorems for CR submanifolds of maximal CR dimension and submanifolds of real codimension two of complex space forms $${\overline{M}}$$, under the algebraic condition on the second fundamental form of the submanifold and the endomorphism induced from the natural almost complex structure of $${\overline{M}}$$ on the tangent bundle of the submanifold.

PSEUDO-PARALLEL KAEHLERIAN SUBMANIFOLDS IN COMPLEX SPACE FORMS

Let $\tilde{M}^{m}(c)$ be a complex $m$-dimensional space form of holomorphic sectional curvature $c$ and $M^{n}$ be a complex $n$-dimensional Kaehlerian submanifold of $\tilde{M}^{m}(c).$ We prove that if $M^{n}$ is pseudo-parallel and $Ln-\frac{1}{2}(n+2)c\geqslant 0$ then $M$ $^{n}$ is totally geodesic. Also, we study Kaehlerian submanifolds of complex space form with recurrent second fundamental form.

Upper bounds for Ricci curvatures for submanifolds in Bochner-Kaehler manifolds

Chen established the relationship between the Ricci curvature and the squared norm of meancurvature vector for submanifolds of Riemannian space form with arbitrary codimension knownas Chen-Ricci inequality. Deng improved the inequality for Lagrangian submanifolds in complexspace form by using algebraic technique. In this paper, we establish the same inequalitiesfor different submanifolds of Bochner-Kaehler manifolds. Moreover, we obtain improvedChen-Ricci inequality for Kaehlerian slant submanifolds of Bochner-Kaehler manifolds.

Ricci curvature on warped product submanifolds of complex space forms and its applications

The upper bound of Ricci curvature conjecture, also known as Chen-Ricci conjecture, was formulated by Chen [B. Y. Chen, Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimension, Glasgow Math. J. 41 (1999) 33–41] and modified by Tripathi [M. M. Tripathi, Improved Chen–Ricci inequality for curvature-like tensors and its applications, Diff. Geom. Appl. 29 (2011) 685–698]. In this paper, first, we define partially minimal isometric immersion of warped product manifolds. Then, we derive a fundamental theorem for Ricci curvature via partially minimal isometric immersions from a warped product pointwise bi-slant submanifolds into complex space forms. Some applications are constructed in terms of Dirichlet energy function, Hamiltonian, Lagrangian and Hessian tensor due to appearance of the positive differential function in the inequality.

Chen inequalities for submanifolds of complex space forms and Sasakian space forms with quarter-symmetric connections

In this paper, we obtain Chen’s inequalities for submanifolds of complex space forms and Sasakian space forms with quarter-symmetric connections.

Biwarped product submanifolds of complex space forms

The class of biwarped product manifolds is a generalized class of product manifolds and a special case of multiply warped product manifolds. In this paper, biwarped product submanifolds of the type [Formula: see text] embedded in the complex space forms are studied. Some characterizing inequalities for the existence of such type of submanifolds are derived. Moreover, we also estimate the squared norm of the second fundamental form in terms of the warping function and the slant function. This inequality generalizes the result obtained by Chen in [B. Y. Chen, Geometry of warped product CR-submanifolds in Kaehler manifolds I, Monatsh. Math. 133 (2001) 177–195]. By the application of derived inequality, we compute the Dirichlet energies of the warping functions involved. A nontrivial example of these warped product submanifolds is provided.

Classification of Casorati ideal Lagrangian submanifolds in complex space forms

In this paper, using optimization methods on Riemannian submanifolds, we establish two improved inequalities for generalized normalized δ-Casorati curvatures of Lagrangian submanifolds in complex space forms. We provide examples showing that these inequalities are the best possible and classify all Casorati ideal Lagrangian submanifolds (in the sense of B.-Y. Chen) in a complex space form. In particular, we generalize the recent results obtained in G.E. Vîlcu (2018) [34].

Classification of warped product pointwise semi-slant submanifolds in complex space forms

The main principle of this paper is to show that, a warped product pointwise semi-slant submanifold of type Mn = Nn1 T xf Nn2? in a complex space form ?M2m (C) admitting shrinking or steady gradient Ricci soliton, whose potential function is a well-define warped function, is an Einstein warped product pointwise semi-slant submanifold under extrinsic restrictions on the second fundamental form inequality attaining the equality in [4]. Moreover, under some geometric assumption, the connected and compactness with nonempty boundary are treated. In this case, we propose a necessary and sufficient condition in terms of Dirichlet energy function which show that a connected, compact warped product pointwise semi-slant submanifold of complex space forms must be a Riemannian product. As more applications, for the first one, we prove that Mn is a trivial compact warped product, when the warping function exist the solution of PDE such as Euler-Lagrange equation. In the second one, by imposing boundary conditions, we derive a necessary and sufficient condition in terms of Ricci curvature, and prove that, a compact warped product pointwise semi-slant submanifold Mn of a complex space form, is either a CR warped product or just a usual Riemannian product manifold. We also discuss some obstructions to these constructions in more details.

An inequality for contact CR-submanifolds in Sasakian space forms

The theory of $$\delta $$ -invariants was initiated by Chen (Arch Math 60:568–578, 1993) in order to find new necessary conditions for a Riemannian manifold to admit a minimal isometric immersion in a Euclidean space. Chen (Int J Math 23(3):1250045, 2012) defined a CR $$\delta $$ -invariant $$\delta (D)$$ for anti-holomorphic submanifolds in complex space forms. Afterwards, Al-Solamy et al. (Taiwan J Math 18:199–217, 2014) established an optimal inequality for this invariant for anti-holomorphic submanifolds in complex space forms. In this article, we prove an optimal inequality for the contact CR $$\delta $$ -invariant on contact CR-submanifolds in Sasakian space forms. An example for the equality case is given.

An optimal inequality for Lagrangian submanifolds in complex space forms involving Casorati curvature

In this paper, we establish an optimal inequality involving normalized δ-Casorati curvature δC(n−1) of Lagrangian submanifolds in n-dimensional complex space forms. We derive a very singular and unexpected result: the lower bounds of the normalized δ-Casorati curvatures δC(n−1) and δCˆ(n−1) in terms of dimension, the holomorphic sectional curvature, the normalized scalar curvature and the squared mean curvature of the submanifold, are different, in contrast to all previous results obtained for several classes of submanifolds in many ambient spaces. We also investigate the equality case of the inequality and prove that a Casorati δC(n−1)-ideal Lagrangian submanifold of a complex space form without totally geodesic points is an H-umbilical Lagrangian submanifold of ratio 4. Some examples are discussed in the last part of the paper, showing that the constants in the inequality obtained in this work are the best possible.