Submanifolds In Sasakian Space Forms Research Articles

Reilly-type inequality for the Φ-Laplace operator on semislant submanifolds of Sasakian space forms

This paper aims to establish new upper bounds for the first positive eigenvalue of the Φ-Laplacian operator on Riemannian manifolds in terms of mean curvature and constant sectional curvature. The first eigenvalue for the Φ-Laplacian operator on closed oriented m-dimensional semislant submanifolds in a Sasakian space form is estimated in various ways. Several Reilly-like inequalities are generalized from our findings for Laplacian to the Φ-Laplacian on semislant submanifolds in a sphere with epsilon =1 and Phi =2.

Classification of Casorati ideal Legendrian submanifolds in Sasakian space forms II

In Lee et al. (2020) [21], the authors of the present article proved two optimal inequalities involving the Casorati curvatures δC(n−1) and δCˆ(n−1) of n-dimensional Legendrian submanifolds in Sasakian space forms and identified the classes of those submanifolds for which the equality cases of both inequalities hold. The aim of this paper is to generalize these results to the case of generalized Casorati curvatures δC(r;n−1) and δCˆ(r;n−1), which are fundamental extrinsic invariants of Riemannian submanifolds originally introduced by Decu et al. (2008) [14] as a natural generalization of δC(n−1) and δCˆ(n−1), where r is any real number such that 0<r<n(n−1) or r>n(n−1), respectively. We also provide examples of submanifolds that are ideal for any given r.

R-Almost Newton–Ricci solitons on Legendrian submanifolds of Sasakian space forms

r-Almost Newton–Ricci solitons on Legendrian submanifolds of Sasakian space forms

Classification of Casorati ideal Legendrian submanifolds in Sasakian space forms

In the first part of this paper, using an optimization method on Riemannian submanifolds, we prove that for any Legendrian submanifold of a Sasakian space form M̄2n+1(c) of constant ϕ-sectional curvature c, we have two sharp inequalities relating some basic extrinsic and intrinsic invariants of the immersion, namely the ϕ-sectional curvature, the normalized scalar curvature, the mean curvature and the δ-Casorati curvatures. In the second part, we classify the family of Casorati ideal Legendrian submanifolds in a Sasakian space form and provide examples supporting the main results of the paper.

Eigenvalue inequalities for the p-Laplacian operator on C-totally real submanifolds in Sasakian space forms

ABSTRACT The main objective of this paper is to provide various estimates of the first eigenvalue of the p-Laplacian operator on a closed oriented n-dimensional C-totally real submanifold in a simply connected Sasakian space form with a constant ϕ-sectional curvature κ. As applications, we generalize the Reilly-type inequality for the Laplacian [Chen and Wei. Reilly-type inequalities for p-Laplacian on submanifolds in space forms. Nonlinear Anal. 2019;84:210–217; Du and Mao. Reilly-type inequalities for p-Laplacian on compact Riemannian manifolds. Front Math China. 2015;10(3):583–594] to the p-Laplacian for C-totally real submanifold in a sphere , for a constant curvature and p = 2.

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.

An Optimal Inequality Related to Characterizations of the Contact Whitney Spheres in Sasakian Space Forms

Let \(N^{2n+1}(c)\) be an \((2n+1)\)-dimensional Sasakian space form with Sasakian structure \((\varphi ,\xi , \eta ,g)\) and constant \(\varphi \)-sectional curvature c. An n-dimensional submanifold \(M^n\) of \(N^{2n+1}(c)\) is called integral if the contact form \(\eta \) restricted to \(M^n\) vanishes. In this paper, we established a general inequality for n-dimensional integral submanifolds in Sasakian space forms involving the norm of the covariant differentiation of both the second fundamental form h and the mean curvature vector field H. Our result is optimal in that we can classify all integral submanifolds realizing the equality case of the inequality. As direct consequence, we give a characterization of the contact Whitney spheres in the Sasakian space forms.

Geometric classification of warped products isometrically immersed into Sasakian space forms

AbstractThe main objective of this paper is to study the warped product pointwise semi‐slant submanifolds which are isometrically immersed into Sasakian manifolds. First, we prove some characterizations results in terms of the shape operator, under which influence a pointwise semi‐slant submanifold of a Sasakian manifold can be reduced to a warped product submanifold. Then, we determine a geometric inequality for the second fundamental form regarding to intrinsic invariant and extrinsic invariant using the Gauss equation instead of the Codazzi equation. Evenmore, we give some applications of this inequality into Sasakian space forms, and we will investigate the status of equalities in the inequality. As a particular case, we provide numerous applications of the Green lemma, the Laplacian of warped functions and some partial differential equations. Some triviality results for connected, compact warped product pointwise semi‐slant submanifolds of Sasakian space form by means of Hamiltonian and the kinetic energy of warped function involving boundary conditions are established.

On the generalized Wintgen inequality for Legendrian submanifolds in Sasakian space forms

The generalized Wintgen inequality was conjectured by De Smet, Dillen, Verstraelen and Vrancken in 1999 for submanifolds in real space forms. It is also known as the DDVV conjecture. It was proven recently by Lu (2011) and by Ge and Tang (2008), independently. The present author established a generalized Wintgen inequality for Lagrangian submanifolds in complex space forms in 2014. In the present paper we obtain the DDVV inequality, also known as generalized Wintgen inequality, for Legendrian submanifolds in Sasakian space forms. Some geometric applications are derived. Also we state such an inequality for contact slant submanifolds in Sasakian space forms.

On pseudo-slant submanifolds of a Sasakian space form

In this paper, we study the geometry of the pseudo-slant submanifolds of a Sasakian space form. Necessary and sufficient conditions are given for a submanifold to be pseudo-slant submanifolds, pseudo-slant product, mixed geodesic and totally geodesic in Sasakian manifolds. Finally, we give some results for totally umbilical pseudo-slant submanifolds of Sasakian manifolds and Sasakian space forms.

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.

Scalar and Ricci curvatures of special contact slant submanifolds in Sasakian space forms

Abstract In this paper we prove a Chen inequality involving the scalar curvature and a Chen-Ricci inequality for special contact slant submanifolds of Sasakian space forms, as the contact versions of the inequalities obtained by the first author in [15] and by both authors in [17], respectively.

Minimal contact CR submanifolds in [formula omitted] satisfying the [formula omitted]-Chen equality

In his book on Pseudo-Riemannian geometry, δ-invariants and applications, B.Y. Chen introduced a sequence of curvature invariants. Each of these invariants is used to obtain a lower bound for the length of the mean curvature vector for an immersion in a real space form. A submanifold is called an ideal submanifold, for that curvature invariant, if and only if it realizes equality at every point. The first such introduced invariant is called δ(2).On the other hand, a well known notion for submanifolds of Sasakian space forms, is the notion of a contact CR-submanifold. In this paper we combine both notions and start the study of minimal contact CR-submanifolds which are δ(2) ideal. We relate this to a special class of surfaces and obtain a complete classification in arbitrary dimensions.

A class of biminimal Legendrian submanifolds in Sasakian space forms

A biminimal submanifold is a critical point of the bienergy functional for any normal variation. We give a complete description of all nonminimal biminimal Legendrian submanifolds in Sasakian space forms which are Legendrian H-umbilical.

An inequality for contact CR-warped product submanifolds of nearly cosymplectic manifolds

Recently, Chen (Monatshefte Math. 133:177-195, 2001) established general sharp inequalities for CR-warped products in a Kaehler manifold. Afterward, Mihai obtained (Geom. Dedic. 109:165-173, 2004) the same inequalities for contact CR-warped product submanifolds of Sasakian space forms and derived some applications. In this paper, we obtain an inequality for the length of the second fundamental form of the warped product submanifold of a nearly cosymplectic manifold in terms of the warping function. The equality case is also discussed.MSC:53C40, 53C42, 53B25.

Lagrangian submanifolds in complex projective space with parallel second fundamental form

From the Riemannian geometric point of view, one of the most fundamental problems in the study of Lagrangian submanifolds is the classiflcation of Lagrangian submanifolds with parallel second fundamental form. In 1980’s, H. Naitoh completely classifled the Lagrangian submanifolds with parallel second fundamental form and without Euclidean factor in complex projective space, by using the theory of Lie groups and symmetric spaces. He showed that such a submanifold is always locally symmetric and is one of the symmetric spaces: SO(k + 1)=SO(k)(k ‚ 2), SU(k)=SO(k)(k ‚ 3), SU(k)(k ‚ 3), SU(2k)=Sp(k)(k ‚ 3), E6=F4. In this paper, we completely classify the Lagrangian submanifolds in complex projective space with parallel second fundamental form by an elementary geometrical method. We prove that such a Lagrangian submanifold is either totally geodesic or the Calabi product of a point with a lower dimensional Lagrangian submanifold with parallel second fundamental form, or the Calabi product of two lower dimensional Lagrangian submanifolds with parallel second fundamental form, or one of the standard symmetric spaces: SU(k)=SO(k)(k ‚ 3), SU(k)(k ‚ 3), SU(2k)=Sp(k)(k ‚ 3), E6=F4. As the arguments are of a local nature, at the same time, due to the correspondence between C-parallel Lagrangian submanifolds in Sasakian space forms and parallel Lagrangian submanifolds in complex space forms, we can also give a complete classiflcation of all Cparallel submanifolds of S 2n+1 equipped with its standard Sasakian structure. 2000 Mathematics Subject Classiflcation: Primary 53B25; Secondary 53C42

Chen-Tripathi Inequality for Warped Product Submanifolds of S-space Forms

We obtain Chen-Tripathi inequality (cf. Theorem 2.1) involving the Laplacian of the warping function and the squared mean curvature of warped product integral submanifolds of an S-space form. Then we find obstructions to the existence of minimal isometric immersions of warped product integral submanifolds of S-space forms. In particular, we get corresponding results for C-totally real warped product submanifolds in Sasakian space forms. We also obtain Chen-Tripathi inequality for warped product submanifolds of an S-space form which are tangent to the structure vector fields of the ambient space.

Improved Chen–Ricci inequality for curvature-like tensors and its applications

We present Chen–Ricci inequality and improved Chen–Ricci inequality for curvature like tensors. Applying our improved Chen–Ricci inequality we study Lagrangian and Kaehlerian slant submanifolds of complex space forms, and C-totally real submanifolds of Sasakian space forms.

Explicit formulas for biharmonic submanifolds in Sasakian space forms

We classify all biharmonic Legendre curves in a Sasakian space form and obtain their explicit parametric equations in the (2n + 1)-dimensional unit sphere endowed with the canonical and deformed Sasakian structures defined by Tanno. We also show that, under the flow-action of the characteristic vector field, a biharmonic integral submanifold becomes a biharmonic anti-invariant submanifold. Then, we obtain new examples of biharmonic submanifolds in the Euclidean sphere S7.

Stability of Biharmonic Legendrian Submanifolds in Sasakian Space Forms

AbstractBiharmonic maps are defined as critical points of the bienergy. Every harmonic map is a stable biharmonic map. In this article, the stability of nonharmonic biharmonic Legendrian submanifolds in Sasakian space forms is discussed.