Structure Vector Field Research Articles

Statistical Killing vector fields on the Hopf hypersurfaces in the complex space forms

Statistical Killing vector field is introduced and Hopf hypersurfaces of complex space forms with the condition of being structural statistical Killing vector field are studied. It is shown that these hypersurfaces have at most three distinct constant principal curvatures.

Chen–Ricci inequality for anti-invariant Riemannian submersions from conformal Kenmotsu space form

AbstractThe aim of this paper is twofold: first, we obtain various curvature inequalities which involve the Ricci and scalar curvatures of horizontal and vertical distributions of anti-invariant Riemannian submersion defined from conformal Kenmotsu space form onto a Riemannian manifold. Second, we obtain the Chen–Ricci inequality for the said Riemannian submersion. The equality cases of all the inequalities are studied. Moreover, these curvature inequalities are studied under two different cases: the structure vector field $$\xi $$ ξ being vertical or horizontal.

Characterizations of warped product pointwise semi-slant submanifolds of Sasakian manifold

A warped product submanifold, whose tangent bundle can be decomposed to two orthogonal distributions; invariant and pointwise slant, is called a warped product pointwise semi-slant submanifold. The objective of this paper is to classify warped product pointwise semi-slant submanifolds, isometrically immersed into a Sasakian manifold. Park [25] provided the (non)-existence of a warped product pointwise semi-slant submanifold in a Sasakian manifold such that the structure vector field is tangential to fibers. In contrast, we provide intriguing theorems on warped product pointwise semi-slant submanifolds in a Sasakian manifold in terms of the shape operator and tensor fields such that the structure vector field is tangential to a base manifold and each fiber is a pointwise slant submanifold.

On statistical submersions from 3-Sasakian statistical manifolds

In this paper, we define and characterize 3-Sasakian statistical manifolds and then investigate statistical submersions from 3-Sasakian statistical manifolds. We prove that invariant statistical submersions from 3-Sasakian statistical manifolds with vertical structure vector fields have 3-Sasakian statistical totally geodesic fibers. Moreover, the base space admits a quaternionic Kähler statistical structure. We construct non-trivial examples to illustrate some results of the paper.

Some Conditions Concerning the Shape Operator of a Real Hypersurface in Complex Projective Space

Let M be a real hypersurface of a complex projective space. For any operator B on M and any nonnull real number k, we can define two tensor fields of type (1,2) on M, B_F^{(k)} and B_T^{(k)}. We will classify real hypersurfaces in complex projective space for which B_F^{(k)} and B_T^{(k)} either take values in the maximal holomorphic distribution mathbb {D} or are parallel to the structure vector field xi , in the particular case of B=A, where A denotes the shape operator of M. We also introduce the concept of A_F^{(k)} and A_T^{(k)} being mathbb {D}-recurrent and classify real hypersurfaces such that either A_F^{(k)} or A_T^{(k)} are mathbb {D}-recurrent.

Another Class of CR-slant Warped Product Submanifolds in Nearly Trans-Sasakian Manifolds

In this paper, we study a CR-slant warped product submanifold of the type B × f Mθ, where Mθ is a slant and B is a CR-product submanifold in a nearly trans-Sasakian manifold. We obtain an inequality for the squared norm of the second fundamental form in two cases, depending on the structure vector field affiliation, in such warped products. The equality case is also considered.

Gradient pseudo‐Ricci solitons of real hypersurfaces

AbstractLet M be a real hypersurface of a complex space form , . Suppose that the structure vector field ξ of M is an eigen vector field of the Ricci tensor S, , β being a function. We study on M, a gradient pseudo‐Ricci soliton () that is an extended concept of gradient Ricci soliton, closely related to pseudo‐Einstein real hypersurfaces. When , we show that M is a Hopf hypersurface.

О почти квазисасакиевых параконтактных структурах на распределениях субримановых многообразий

The concept of an almost quasi-Sasakian paracontact structure is introduced. In contrast to the quasi-Sasakian paracontact structure, the introduced structure does not have the normality property. The invariants of the intrinsic geometry of an almost quasi-Sasakian paracontact structure are studied. The intrinsic geometry of an almost paracontact metric manifold is understood to mean those of its geometric properties that depend only on the parallel translation determined by the intrinsic connection. In particular, it is proved that the vanishing of the Schouten tensor implies that the structure vector field is a Killing vecto r field. An example of an almost quasi-Sasakian paracontact structure is given. Almost quasi-Sasakian paracontact structures arise naturally on distributions of sub-Riemannian manifolds. On a non-involutive distribution D of a sub-Riemannian manifold M of contact type, an almost paracontact metric structure is defined, which is called an extended structure. It is proved that on a distribution of zero curvature, the extended structure is an almost quasi -Sasakian paracontact structure. A characteristic property of a distribution of zero curvature is the vanishing of the Schouten curvature tensor. The well-known classification of extended structures is analyzed based on the properties of the fundamental tensor F of type (0, 3) associated to the extended structure. In accordance with this classification, there are 212 classes of almost paracontact metric structure, among which there are 12 basic ones. A class containing an extended almost quasi-Sasakian paracontact structure is found.

Seismic metamaterials based on coupling mechanism of inertial amplification and local resonance

Inertial amplification mechanisms could be used to control the propagation of elastic waves in beams and slabs, but it was a difficult problem to apply inertial amplification mechanisms to seismic metamaterials to design low-frequency broadband. This paper presents a inertially amplified locally resonant seismic metamaterial (IALR-SM) using the coupling mechanism of inertial amplification and local resonance. In contrast to the locally resonant seismic metamaterial (LRSM), the large-mass columns as local resonators of IALR-SM are attached to the connector and small-mass columns to form the inertia amplification structures. The finite element method calculates the eigenmodes’ band structure and displacement vector field. Numerical results indicated that the bandwidth of the IALR-SM increases by 194% compared to LRSM for seismic surface waves below 20 Hz. The formation mechanism of the yield in large band gaps is attributed to the coupling mechanism of local resonance and inertial amplification. In addition, the numerical effects of geometric parameters on the band gaps are investigated. The findings showed that the side length of the small-mass columns plays a vital role in determining which coupling mechanism is dominant. Finally, field experiments demonstrated that the IALR-SM generates low-frequency broadband.

К геометрии субримановых многообразий, оснащенных канонической четверть-симметрической связностью

In this article, a sub-Riemannian manifold of contact type is under­stood as a Riemannian manifold equipped with a regular distribution of codimension-one and by a unit structure vector field orthogonal to this distribution. This vector field is called a structural. On a sub-Riemannian manifold of contact type, a quarter-symmetric connection is defined, which is associated with an endomorphism that preserves the distribution of the sub-Riemannian manifold. It is proved that if the connection under study is metric, then the endomorphism associated to it is uniquely de­fined. The structure of the associated endomorphism is found. In the case when the structure vector field is a field of infinitesimal isometries, the quarter-symmetric connection is called the canonical N-connection. An expression is found for the curvature tensor of the canonical N-connection in terms of the Riemann curvature tensor. The properties of the Schouten curvature tensor are investigated, which provide, in particular, the neces­sary symmetries of the curvature tensor of an N-connection for its sec­tional curvature to be well-defined. A relation between the sectional cur­vature of the canonical N-connection and the sectional curvature of the Levi-Civita connection is found. Necessary and sufficient conditions are found under which the sectional curvature of the N-connection and the sectional curvature of the Levi-Civita connection coincide.

On contact screen conformal null submanifolds

First, we prove that indefinite Sasakian manifolds do not admit any screen conformal $r$-null submanifolds, tangent to the structure vector field. We, therefore, define a special class of null submanifolds, called; {\it contact screen conformal} $r$-null submanifold of indefinite Sasakian manifolds. Several characterization results, on the above class of null submanifolds, are proved. In particular, we prove that such null submanifolds exist in indefinite Sasakian space forms of constant holomorphic sectional curvatures of $-3$.

Real hypersurfaces in ℂP 2 and ℂH 2 with constant scalar curvature

Abstract In this paper, Hopf hypersurfaces in a complex projective plane ℂP 2(c) or a complex hyperbolic plane ℂH 2(c) with constant scalar curvature are classified. For a non-Hopf hypersurface in ℂP 2(c) with constant scalar curvature r, it is proved that if the structure vector field is an eigenvector of the Ricci operator, then either r = 7c/2 or r = 3c/2. Moreover, these two cases are determined completely under an additional condition.

Trans-Sasakian 3-manifolds with Einstein-like Ricci operators

Let M be a trans-Sasakian 3-manifold such that the Ricci curvature of the structure vector field vanishes. In this paper, it is proved that if M is compact, then it is homothetic to a cosymplectic manifold. Without the compactness assumption, we prove that M is locally isometric to the product of ℝ and a Kahler surface of constant curvature if the Ricci operator of M is of Codazzi type or cyclic parallel. We classify trans-Sasakian 3-manifolds with Killing type Ricci tensors. We construct some concrete examples to illustrate the above theorems.

Hypersurfaces of a Sasakian manifold - revisited

We study orientable hypersurfaces in a Sasakian manifold. The structure vector field ξ of a Sasakian manifold determines a vector field v on a hypersurface that is the component of the Reeb vector field ξ tangential to the hypersurface, and it also gives rise to a smooth function σ on the hypersurface, namely the projection of ξ on the unit normal vector field N. Moreover, we have a second vector field tangent to the hypersurface, given by mathbf{u}=-varphi (N). In this paper, we first find a necessary and sufficient condition for a compact orientable hypersurface to be totally umbilical. Then, with the assumption that the vector field u is an eigenvector of the Laplace operator, we find a necessary condition for a compact orientable hypersurface to be isometric to a sphere. It is shown that the converse of this result holds, provided that the Sasakian manifold is the odd dimensional sphere mathbf{S}^{2n+1}. Similar results are obtained for the vector field v under the hypothesis that this is an eigenvector of the Laplace operator. Also, we use a bound on the integral of the Ricci curvature Ric ( mathbf{u},mathbf{u} ) of the compact hypersurface to find a necessary condition for the hypersurface to be isometric to a sphere. We show that this condition is also sufficient if the Sasakian manifold is mathbf{S}^{2n+1}.

Многообразия Кенмоцу с нулевым тензором Риччи – Схоутена

The paper is dedicated to the investigation of the interior geometry of the Kenmotsu manifolds M. By the interior geometry of the manifold M we mean the aggregate of the properties of the manifold that depend only on the closing of the distribution D of the Kenmotsu manifold as well as on the parallel transport of the vectors from the distribution D along arbitrary curves of the manifold. The invariants of the interior geometry of a Kenmotsu manifold are the following: the Schouten curvature tensor; the 1-form η generating the distribution D; the Lie derivative of the metric tensor g along the structure vector field ; the Schouten-Wagner admissible tensor fields with the components with respect to adapted coordinates; the structural endomorphism φ; the endomorphism N that allows to prolong the interior connection to a connection in a vector bundle. A special attention is payed to the Ricci-Schouten tensor. In particular, it is stated that a Kenmotsu manifold with zero Ricci-Schouten tensor is an Einstein manifold. Conversely, if M is an η-Einstein Kenmotsu manifold and then M is an Einstein manifold with zero Ricci-Schouten tensor. It is proved that the Ricci-Schouten tensor is zero if and only if the Kenmotsu manifold M is locally Ricci-symmetric. This implies the following well-known result: a Kenmotsu manifold is an Einstein manifold if and only if it is locally Ricci-symmetric. An N-connection with torsion, is introduced; this connection is Ricci-flat if and only if M is an Einstein manifold.

Five-Dimensional Contact CR-Submanifolds in S 7 ( 1 )

Due to the remarkable property of the seven-dimensional unit sphere to be a Sasakian manifold with the almost contact structure (φ,ξ,η), we study its five-dimensional contact CR-submanifolds, which are the analogue of CR-submanifolds in (almost) Kählerian manifolds. In the case when the structure vector field ξ is tangent to M, the tangent bundle of contact CR-submanifold M can be decomposed as T(M)=H(M)⊕E(M)⊕Rξ, where H(M) is invariant and E(M) is anti-invariant with respect to φ. On this occasion we obtain a complete classification of five-dimensional proper contact CR-submanifolds in S7(1) whose second fundamental form restricted to H(M) and E(M) vanishes identically and we prove that they can be decomposed as (multiply) warped products of spheres.

3-Dimensional real hypersurfaces with transversal $\eta$-Killing Ricci tensor field of a complex space form

We classify real hypersurfaces of a non-flat complex space form of complex dimension 2 satisfying the condition that the Ricci tensor S is the transversal Killing tensor field on the direction of the structure vector field $\xi$ and $S\xi = \beta\xi$ where $\beta$ is a function.

Remarks on Riemann and Ricci Solitons in $(\alpha,\beta)$-Contact Metric Manifolds

We study almost Riemann solitons and almost Ricci solitons in an $(\alpha,\beta)$-contact metric manifold satisfying some Ricci symmetry conditions, treating the case when the potential vector field of the soliton is pointwise collinear with the structure vector field.

Lightlike Hypersurfaces of an Indefinite Para Sasakian Manifold

In this paper, we initiate the study of lightlike hypersurfaces of an indefinite almost paracontact metric manifold which are tangent to the structure vector field. In particular, we give definitions of invariant lightlike hypersurfaces and screen semi-invariant lightlike hypersurfaces, and give some examples. Integrability conditions for the distributions involved in the screen semi-invariant lightlike hypersurface of an indefinite para Sasakian manifold are investigated.

Associated Families of Surfaces in Warped Products and Homogeneous Spaces

We classify Riemannian surfaces admitting associated families in three dimensional homogeneous spaces with four-dimensional isometry groups and in a wide family of (semi-Riemannian) warped products, with an extra natural condition (namely, rotating structure vector field). We prove that, provided the surface is not totally umbilical, such families exist in both cases if, and only if, the ambient manifold is a product and the surface is minimal. In particular, there exists no associated families of surfaces with rotating structure vector field in the Heisenberg group.