Sasakian Space Forms Research Articles

Hemi-slant ξ⊥-Riemannian submersions in contact geometry

M. A. Akyol and R. Sar? [On semi-slant ??-Riemannian submersions, Mediterr. J. Math. 14(6) (2017) 234.] defined semi-slant ??-Riemannian submersions from Sasakian manifolds onto Riemannian manifolds. As a generalization of the above notion and natural generalization of anti-invariant ??-Riemannian submersions, semi-invariant ??-Riemannian submersions and slant submersions, we study hemi-slant ??-Riemannian submersions from Sasakian manifolds onto Riemannian manifolds. We obtain the geometry of foliations, give some examples and find necessary and sufficient condition for the base manifold to be a locally product manifold. Moreover, we obtain some curvature relations from Sasakian space forms between the total space, the base space and the fibres.

Chen optimal inequalities of CR-warped products of generalized Sasakian space form

Our main objective of this paper is to derive the relationship between the main extrinsic invariant, and the contact CR δ-invariant (new intrinsic invariant) on a generic submanifold in trans-Sasakian generalized Sasakian space forms. Further, we find a lower bound of the squared norm of the mean curvature (main extrinsic invariant) in terms of a CR δ-intrinsic invariant, and the Laplacian of the warping function for CR-warped products in the same ambient space forms. We also investigate the classifications and triviality of connected, compact CR-warped product manifolds isometrically immersed into the trans-Sasakian generalized Sasakian space forms.

Ricci curvature of contact CR-warped product submanifolds in generalized Sasakian space forms admitting nearly Sasakian structure

<abstract> <p>The objective of this paper is to achieve the inequality for Ricci curvature of a contact CR-warped product submanifold isometrically immersed in a generalized Sasakian space form admitting a nearly Sasakian structure in the expressions of the squared norm of mean curvature vector and warping function. In addition, the equality case is likewise discussed. 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>

Some Results on Generalized Sasakian Space Forms

Abstract In the present frame work, we studied the semi generalized recurrent, semi generalized ϕ-recurrent, extended generalized ϕ-recurrent and concircularly locally ϕ-symmetric on generalized Sasakian space forms.

F-Biharmonic Submanifolds of Generalized Space Forms

We study f-biharmonic in both generalized complex and Sasakian space forms. We prove necessary and sufficient condition for f-biharmonicity in the general case and many particular cases. Some non-existence results are also obtained.

On interpolating sesqui-harmonic Legendre curves in Sasakian space forms

We consider interpolating sesqui-harmonic Legendre curves in Sasakian space forms. We find the necessary and sufficient conditions for Legendre curves in Sasakian space forms to be interpolating sesqui-harmonic. Finally, we obtain a proper example for an interpolating sesqui-harmonic Legendre curve in a Sasakian space form.

A Closed Form for Slant Submanifolds of Generalized Sasakian Space Forms

The Maslov form is a closed form for a Lagrangian submanifold of C m , and it is a conformal form if and only if M satisfies the equality case of a natural inequality between the norm of the mean curvature and the scalar curvature, and it happens if and only if the second fundamental form satisfies a certain relation. In a previous paper we presented a natural inequality between the norm of the mean curvature and the scalar curvature of slant submanifolds of generalized Sasakian space forms, characterizing the equality case by certain expression of the second fundamental form. In this paper, first, we present an adapted form for slant submanifolds of a generalized Sasakian space form, similar to the Maslov form, that is always closed. And, in the equality case, we studied under which circumstances the given closed form is also conformal.

Magnetic cubics in Riemannian manifolds associated with different magnetic fields

Analogous to magnetic curves generalizing geodesics, magnetic cubics are generalized Riemannian cubics in the presence of magnetic fields. In this paper, we mainly study magnetic cubics in Riemannian manifolds associated with different magnetic fields. In addition to presenting the differential equations for magnetic cubics, we find the connection between magnetic cubics and Riemannian cubics in Lie groups and Sasakian space forms. Furthermore, some closed form solutions are achieved for special magnetic cubics, for instance, the so-called null magnetic cubics.

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.

Symmetry properties of complex contact space form

It is well known that a Sasakian space form is pseudo-symmetric [M. Belkhelfa, R. Deszcz and L. Verstraelen, Symmetry properties of Sasakian space-forms, Soochow J. Math. 31(4) (2005) 611–616], therefore it is Ricci-pseudo-symmetric. In this paper, we proved that a normal complex contact manifold is Ricci-semi-symmetric if and only if it is an Einstein manifold; moreover, we showed that a complex contact space form [Formula: see text] with constant [Formula: see text]-sectional curvature [Formula: see text] is properly Ricci-pseudo-symmetric [Formula: see text] if and only if [Formula: see text]; in this case [Formula: see text]. We gave an example of properly Ricci-pseudo-symmetric complex contact space form. On the other hand, we proved the non-existence of proper pseudo-symmetric ([Formula: see text]) complex contact space form [Formula: see text]

Curvature of $$C_5\oplus C_{12}$$-Manifolds

The Chinea–Gonzalez class $$C_5\oplus C_{12}$$ consists of the almost contact metric manifolds that are locally described as double-twisted product manifolds $$I\times _{(\lambda _1,\lambda _2)}{\widehat{M}}$$ , $$I\subset {\mathbb {R}}$$ being an open interval, $${\widehat{M}}$$ a Kahler manifold and $$\lambda _1,\lambda _2$$ smooth positive functions. In this article, we investigate the behavior of the curvature of $$C_5\oplus C_{12}$$ -manifolds. Particular attention to the N(k)-nullity condition is given and some local classification theorems in dimension $$2n+1\ge 5$$ are stated. This allows us to classify $$C_5\oplus C_{12}$$ -manifolds that are generalized Sasakian space forms. In addition, we provide explicit examples of these 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.

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.

Contact CR-warped product submanifolds of a generalized Sasakian space form admitting a nearly Sasakian structure

Contact CR-warped product submanifolds of a generalized Sasakian space form admitting a nearly Sasakian structure

Classification of biharmonic $\mathcal{C}$-parallel Legendrian submanifolds in 7-dimensional Sasakian space forms

In [5], D. Fetcu and C. Oniciuc presented the classification result for biharmonic $\mathcal{C}$-parallel Legendrian submanifolds in 7-dimensional Sasakian space forms. However, it is incomplete. In this paper, all such submanifolds are explicitly determined.

Strongly pseudo-convex CR space forms

AbstractFor a contact manifold, we study a strongly pseudo-convex CR space form with constant holomorphic sectional curvature for the Tanaka-Webster connection. We prove that a strongly pseudo-convex CR space formMis weakly locally pseudo-Hermitian symmetric if and only if (i) dimM= 3, (ii)Mis a Sasakian space form, or (iii)Mis locally isometric to the unit tangent sphere bundleT1(𝔿n+1) of a hyperbolic space 𝔿n+1of constant curvature −1.

F-biharmonic integral submanifolds in generalized sasakian space forms

We study f-biharmonic integral submanifolds and integral C-parallel submanifolds in generalized Sasakian space forms. As an application, we find the f-biharmonicity conditions for the integral and integral C-parallel submanifolds in Sasakian, ?-Sasakian, Kenmotsu and cosymplectic space forms. Finally, we give also some examples of f-biharmonic integral submanifolds in Sasakian space forms.

Pseudosymmetry properties of generalised Wintgen ideal Legendrian submanifolds

For Legendrian submanifolds Mn in Sasakian space forms ?M2n+1(c), I. Mihai obtained an inequality relating the normalised scalar curvature (intrinsic invariant) and the squared mean curvature and the normalised scalar normal curvature of M in the ambient space ?M (extrinsic invariants) which is called the generalised Wintgen inequality, characterising also the corresponding equality case. And a Legendrian submanifold Mn in Sasakian space forms ?M2n+1(c) is said to be generalised Wintgen ideal Legendrian submanifold of ?M2n+1(c) when it realises at everyone of its points the equality in such inequality. Characterisations based on some basic intrinsic symmetries involving the Riemann-Cristoffel curvature tensor, the Ricci tensor and the Weyl conformal curvature tensor belonging to the class of pseudosymmetries in the sense of Deszcz of such generalised Wintgen ideal Legendrian submanifolds are given.

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.

Optimal casorati inequalities on bi-slant submanifolds of generalized sasakian space forms

In this paper, we use T Oprea's optimization method to establish some optimal Casorati inequalities, which involve the normalized scalar curvature for bi-slant submanifolds of generalized Sasakian space forms. In the continuation, we show that in both cases, the equalities hold if and only if submanifolds are invariantly quasi-umbilical.