Contact CR-submanifolds Research Articles

Contact CR $ \delta $-invariant: an optimal estimate for Sasakian statistical manifolds

<p>Chen (1993) developed the theory of $ \delta $-invariants to establish novel necessary conditions for a Riemannian manifold to allow a minimal isometric immersion into Euclidean space. Later, Siddiqui et al. (2024) derived optimal inequalities involving the CR $ \delta $-invariant for a generic statistical submanifold in a holomorphic statistical manifold of constant holomorphic sectional curvature. In this work, we extend the study of such optimal inequality to the contact CR $ \delta $-invariant on contact CR-submanifolds in Sasakian statistical manifolds of constant $ \phi $-sectional curvature. This paper concludes with a summary and final remarks.</p>

Solitonic View of Generic Contact CR-Submanifolds of Sasakian Manifolds with Concurrent Vector Fields

This paper mainly devotes to the study of some solitons such as Ricci and Yamabe solitons and also their combination called Ricci-Yamabe solitons. In the geometry of solitons, a fundamental question is to identify the conditions under which these solitons can be trivial. Firstly, in this paper we study some extensive results on generic contact CR-submanifolds of Sasakian manifolds endowed with concurrent vector fields. Then some applications of solitons such as Ricci and Ricci-Yamabe solitons on such submanifolds with concurrent vector fields in the same ambient manifold have been discussed.

On Contact CR-Submanifold of a Kenmotsu Manifold with Killing Tensor Field

The object of this paper is to study the Contact CR-submanifold of a Kenmotsu manifold with the help of a killing tensor field and deduce some results.

A Topological sphere theorem for contact CR-warped product submanifolds of an odd-dimensional unit sphere

AbstractIn this paper, we investigate topological sphere theorems for compact minimal contact CR-submanifolds of odd dimensional unit sphere. We show that if an inequality involving the warping function and the scalar curvature of the fibers is satisfied, a compact minimal contact CR-warped product submanifold of the odd dimensional unit sphere is homeomorphic to the sphere. In particular case, for 5-dimensional unit sphere, we show that a 4-dimensional compact minimal contact CR-warped product submanifold is homeomorphic to a sphere if ∣∇lnf∣2 < 1 is satisfied. By using Bonnet-Myers’s theorem we give a result about fundamental group and by Leung’s theorem we obtain a result about homology groups of a contact CR-warped submanifold.

Contact CR-submanifolds of trans-Sasakian manifolds with respect to quarter symmetric non-metric connection

The present paper deals with the study of contact CR-submanifolds of trans-Sasakian manifolds with respect to quarter symmetric non-metric connection. We investigate totally geodesic leaves and integrability of the distributions and also study the totally umbilical contact CR-submanifolds of trans-Sasakian manifolds. At last we give an example to verify a relation.

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.

CR Maximal Dimensional Submanifolds of Kenmotsu Space Forms

For n > 5, we investigate (n + 1)-dimensional contact CR-submanifolds M of (n − 1) contact CR-dimension in a complete simply connected Kenmotsu space form of constant ϕ-holomorphic sectional curvature c which satisfy the condition h(FX,Y ) + h(X,FY ) = 0 for any vector fields X,Y tangent to M, where h and F denote the second fundamental form and a skew-symmetric endomorphism acting on the tangent space of M, respectively.

Homology of contact CR-submanifolds

In this paper, homology of a contact CR-submanifold of a real hypersurface, which has naturally almost contact metric structure induced from the complex Euclidean space Cm, is examined. More precisely, nonexistence of stable integral currents on a compact contact CR-submanifold of real hypersurface of canonical complex space form Cm is investigated and vanishing theorems concerning the homology groups are obtained.

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.

A class of warped product submanifolds of Kenmotsu manifolds

In 2018 Naghi et al. studied warped product skew CR-submanifold of the form M = M1×f M⊥ of a Kenmotsu manifold M¯ throughout the paper , where M1 = MT ×Mθ and MT , M⊥, Mθ represents invariant, antiinvariant, proper slant submanifold of M¯ . Next, in 2019 Hui et al. studied another class of warped product skew CR-submanifold of the form M = M2 ×f MT of M¯ , where M2 = M⊥ × Mθ . The present paper deals with the study of a class of warped product submanifold of the form M = M3 ×f Mθ of M¯ , where M3 = MT × M⊥ and MT , M⊥, Mθ represents invariant, antiinvariant and proper pointwise slant submanifold of M¯ . A characterization is given on the existence of such warped product submanifolds which generalizes the characterization of warped product contact CR-submanifolds of the form M⊥ ×f MT , studied by Uddin et al. in 2017 and also generalizes the characterization of warped product semi-slant submanifolds of the form MT ×f Mθ , studied by Uddin in the same year. Beside that some inequalities on the squared norm of the second fundamental form are obtained which are also generalizations of the inequalities obtained in the just above two mentioned papers respectively.

Contact CR Submanifolds of maximal Contact CR dimension of Sasakian Space Form

In this paper, we investigate contact CR submanifolds of contact CR dimension in Sasakian space form and introduce the general structure of these submanifolds and then studying structures of this submanifols with the condition  h(FX,Y)+h(X,FY)=g(FX,Y)zeta, for the normal vector field zeta, which is nonzero, and we classify these submanifolds../files/site1/files/61/0Abstract.pdf

On some geometric properties of CR-submanifolds of a Sasakian manifold

In this paper, we study the geometry of the contact CR-submanifolds of a Sasakian manifold. By searching the parallelism conditions of the tensors reduced in the submanifolds, some properties and results obtained from the definition of the contact CR-submanifolds are given. Also, necessary and sufficient conditions are given for a submanifold to be a contact CR-submanifold, contact CR-product, D, D⊥ and mixed totally geodesic in a Sasakian manifold.

Submersions of Contact CR-Submanifolds of Generalized Quasi Sasakian Manifolds

In this paper, we discuss some properties of almost contact metric submersion of contact CR-submanifolds of generalized quasi-Sasakian manifold and derive some results based on their curvatures’s differential geometry. We also study de-Rham cohomology of CR-submanifold of generalized quasi-Sasakian manifold under the submersion.

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.

On contact CR-submanifolds of a cosymplectic manifold

In this paper, we study the differential geometry of contact CR-submanifolds of a cosymplectic manifold. Necessary and sufficient conditions are given for a submanifold to be a contact CR-submanifold in cosymplectic manifolds and cosymplectic space forms. Finally, the induced structures on submanifolds are investigated, these structures are categorized and we discuss these results.

EXISTENCE OF PROPER CONTACT CR PRODUCT AND MIXED FOLIATE CONTACT CR SUBMANIFOLDS OF E2m+1(-3)

The first purpose of this paper is to study contact CR submanifolds of Sasakian manifolds and investigate some properties concernig with <TEX>${\phi}$</TEX>-holomorphic bisectional curvature. The second purpose is to show an existence theorem of mixed foliate proper contact CR submanifolds in the standard Sasakian space form <TEX>$E^{2m+1}$</TEX>(-3) with constant <TEX>${\phi}$</TEX>-sectional curvature -3.

CERTAIN CLASS OF CONTACT CR-SUBMANIFOLDS OF A SASAKIAN SPACE FORM

In this paper we investigate (n+1)(<TEX>$n{\geq}3$</TEX>)-dimensional contact CR-submanifolds M of (n-1) contact CR-dimension in a complete simply connected Sasakian space form of constant <TEX>${\phi}$</TEX>-holomorphic sectional curvature <TEX>$c{\neq}-3$</TEX> which satisfy the condition h(FX, Y)+h(X, FY) = 0 for any vector fields X, Y tangent to M, where h and F denote the second fundamental form and a skew-symmetric endomorphism (defined by (2.3)) acting on tangent space of M, respectively.

Horizontally submersions of contact $CR$-submanifolds

In this paper, we discuss some geometric properties of almost contact metric submersions involving symplectic manifolds. We show that the structures of quasi-K-cosymplectic and quasi-Kenmotsu manifolds are related to (1, 2)-symplectic structures. For horizontally submersions of contact CR-submanifolds of quasi-K-cosymplectic and quasi-Kenmotsu manifolds, we study the principal characteristics and prove that their total spaces are CR-product. Curvature properties between curvatures of quasi-K-cosymplectic and quasi-Kenmotsu manifolds and the base spaces of such submersions are also established. We finally prove that, under a certain condition, the contact CR-submanifold of a quasi Kenmotsu manifold is locally a product of a totally geodesic leaf of an integrable horizontal distribution and a curve tangent to the normal distribution.

Contact CR-submanifolds of an indefinite Lorentzian para-Sasakian manifold

Abstract In this paper we prove some properties of the indefinite Lorentzian para-Sasakian manifolds. Section 1 is introductory. In Section 2 we define D-totally geodesic and D⊥-totally geodesic contact CRsubmanifolds of an indefinite Lorentzian para-Sasakian manifold and deduce some results concerning such a manifold. In Section 3 we state and prove some results on mixed totally geodesic contact CR-submanifolds of an indefinite Lorentzian para-Sasakian manifold. Finally, in Section 4 we obtain a result on the anti-invariant distribution of totally umbilic contact CR-submanifolds of an indefinite Lorentzian para-Sasakian manifold.

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.