Sasakian Manifolds Research Articles

\eta-RICCI SOLITONS IN LORENTZIAN \alpha-SASAKIAN MANIFOLDS

In the present paper, we have studied $\eta$-Ricci solitons in Lorentzian \\ $\alpha-$Sasakian manifolds satisfying certain curvature conditions. The existence of $\eta-$Ricci soliton in a Lorentzian $\alpha-$Sasakian manifold has been proved by a concrete example.

English

We characterize generalized Douglas-Weyl Randers metrics in terms of their Zermelo navigation data. Then, we study the Randers metrics induced by some important classes of almost contact metrics. Furthermore, we construct a family of generalized Douglas-Weyl Randers metrics which are not R-quadratic. We show that the Randers metric induced by a Kenmotsu manifold is a Douglas metric which is not of isotropic S-curvature. We show that the Randers metric induced by a Kenmotsu or Sasakian manifold is not Einsteinian. By using D-homothetic deformation of a Kenmotsu or Sasakian manifold, we construct a family of generalized Douglas-Weyl Randers metrics and show that the Lie group of projective transformations does not act transitively on the set of generalized Douglas-Weyl Randers metrics.

Submanifolds of a conformal Sasakian manifold

In the present paper, some results on geometry of conformal Sasakian manifolds and their associated submanifolds are provided. Besides this an example of a three-dimensional conformal Sasakian manifold is constructed for illustration that is not Sasakian manifold.

Invariance of basic Hodge numbers under deformations of Sasakian manifolds

We show that the Hodge numbers of Sasakian manifolds are invariant under arbitrary deformations of the Sasakian structure. We also present an upper semi-continuity theorem for the dimensions of kernels of a smooth family of transversely elliptic operators on manifolds with homologically orientable transversely Riemannian foliations. We use this to prove that the partial {bar{partial }}-lemma and being transversely Kähler are rigid properties under small deformations of the transversely holomorphic structure which preserve the foliation. We study an example which shows that this is not the case for arbitrary deformations of the transversely holomorphic foliation. Finally we point out an application of the upper-semi continuity theorem to K-contact manifolds.

On complex Sasakian manifolds

In this article we study a class of normal{\theta}complex{\theta}contact{\theta}metric{\theta}manifold which is called a complex Sasakian manifold. This kind of manifold has a globally defined complex contact form and normal complex contact structure. We give the definition of a complex Sasakian manifold by consider the real case and we present general properties. Also we obtain some useful curvature relations. Finally we examine flatness conditions for general curvature tensor B.

On the p-pseudoharmonic map heat flow

In this paper, we consider the heat flow for [Formula: see text]-pseudoharmonic maps from a closed Sasakian manifold [Formula: see text] into a compact Riemannian manifold [Formula: see text]. We prove global existence and asymptotic convergence of the solution for the [Formula: see text]-pseudoharmonic map heat flow, provided that the sectional curvature of the target manifold [Formula: see text] is non-positive. Moreover, without the curvature assumption on the target manifold, we obtain global existence and asymptotic convergence of the [Formula: see text]-pseudoharmonic map heat flow as well when its initial [Formula: see text]-energy is sufficiently small.

On quasi bi-slant submersions from Sasakian manifolds onto Riemannian manifolds

In this paper, we study the quasi bi-slant submersions from Sasakian manifolds onto Riemannian manifolds. We define quasi bi-slant submersions from almost contact metric manifolds onto Riemannian manifolds which are generalization of hemi-slant submersions and semi-slant submersions with some examples. We also discuss the geometry of leaves of distributions which are involved in the definition of this submersion and obtain coditions for such submersions to be integrable and totally geodesic. Further, we give a characterization theorem for proper quasi bi-slant submersions with totally umbilical fibres.

$LP$-Sasakian manifolds with generalized symmetric metric connection

$LP$-Sasakian manifolds with generalized symmetric metric connection

HOLONOMY AND 3-SASAKIAN HOMOGENEOUS MANIFOLDS VERSUS SYMPLECTIC TRIPLE SYSTEMS

Our aim is to support the choice of two remarkable connections with torsion in a 3-Sasakian manifold, proving that, in contrast to the Levi-Civita connection, the holonomy group in the homogeneous cases reduces to a proper subgroup of the special orthogonal group, of dimension considerably smaller. We realize the computations of the holonomies in a unified way, by using as a main algebraic tool a nonassociative structure, that of a symplectic triple system.

Screen Generic Lightlike Submanifolds of Indefinite Sasakian Manifolds

In this paper, we introduce the general notion of screen generic lightlike submanifolds of indefinite Sasakian manifolds. We study screen generic lightlike submnaifolds, contact totally umbilical screen generic lightlike submanifolds, and minimal screen generic lightlike submanifolds of indefinite Sasakian manifolds. Also, we provide some examples of screen generic lightlike submanifold and minimal screen generic lightlike submanifold of an indefinite Sasakian 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.

On warped product semi invariant submanifolds of nearly (ε,δ)-trans Sasakian manifold

In this paper, we have concentrated on the inquest of warped product semi-invariant submanifolds of a nearly (ε, δ)-trans-Sasakian manifold. Firstly, some properties of this structure are acquired. Further, we established the warped product of the type E⊥ x yEP is a usual Riemannian product of E⊥ and EP , where E⊥ and EP are anti-invariant and invariant submanifolds of a nearly (ε, δ)-trans-Sasakian manifold M , respectively. Also, we explored the warped product of the type EP x yE⊥ and acquired a depiction for such type of warped product.

∗-η-Ricci Soliton within the Framework of Sasakian Manifold

In this paper we study ∗-η-Ricci soliton on Sasakian manifolds. Here, we have discussed some curvature properties on Sasakian manifold admitting ∗-η-Ricci soliton. We have obtained some significant results on ∗-η-Ricci soliton in Sasakian manifolds satisfying R(ξ, X) · S = 0, S(ξ, X) · R = 0, P̄ (ξ, X) · S = 0, where P̄ is Pseudo-projective curvature tensor.The conditions for ∗-η-Ricci soliton on ϕ-conharmonically flat and ϕ-projectively flat Sasakian manifolds have been obtained in this article. Lastly, we have given an example of 5-dimensional Sasakian manifolds satisfying ∗-η-Ricci soliton.

Pointwise Slant and Pointwise Semi-Slant Submanifolds in Almost Contact Metric Manifolds

In almost contact metric manifolds, we consider two kinds of submanifolds: pointwise slant, pointwise semi-slant. On these submanifolds of cosymplectic, Sasakian and Kenmotsu manifolds, we obtain characterizations and study their topological properties and distributions. We also give their examples. In particular, we obtain some inequalities consisting of a second fundamental form, a warping function and a semi-slant function.

Hypersurfaces of a Sasakian Manifold

We extend the study of orientable hypersurfaces in a Sasakian manifold initiated by Watanabe. The Reeb vector field ξ of the Sasakian manifold induces a vector field ξ T on the hypersurface, namely the tangential component of ξ to hypersurface, and it also gives a smooth function ρ on the hypersurface, which is the projection of the Reeb vector field on the unit normal. First, we find volume estimates for a compact orientable hypersurface and then we use them to find an upper bound of the first nonzero eigenvalue of the Laplace operator on the hypersurface, showing that if the equality holds then the hypersurface is isometric to a certain sphere. Also, we use a bound on the energy of the vector field ∇ ρ on a compact orientable hypersurface in a Sasakian manifold in order to find another geometric condition (in terms of mean curvature and integral curves of ξ T ) under which the hypersurface is isometric to a sphere. Finally, we study compact orientable hypersurfaces with constant mean curvature in a Sasakian manifold and find a sharp upper bound on the first nonzero eigenvalue of the Laplace operator on the hypersurface. In particular, we show that this upper bound is attained if and only if the hypersurface is isometric to a sphere, provided that the Ricci curvature of the hypersurface along ∇ ρ has a certain lower bound.

7D supersymmetric Yang-Mills on hypertoric 3-Sasakian manifolds

We study 7D maximally supersymmetric Yang-Mills theory on 3-Sasakian manifolds. For manifolds whose hyper-Kähler cones are hypertoric we derive the perturbative part of the partition function. The answer involves a special function that counts integer lattice points in a rational convex polyhedral cone determined by hypertoric data. This also gives a more geometric structure to previous enumeration results of holomorphic functions in the literature. Based on physics intuition, we provide a factorisation result for such functions. The full proof of this factorisation using index calculations will be detailed in a forthcoming paper.

ON NON-INVARIANT HYPERSURFACES OF AN ε-PARA SASAKIAN MANIFOLD

In the present paper non-invariant hypersurfaces of an ε- para Sasakian manifold of an induced structure (f,g,u,v,λ) are studied. Some properties followed by this structure are obtained. A necessary and sufficient condition for totally umbilical non-invariant hypersurfaces equipped with (f,g,u,v,λ)- structure of ε-para Sasakian manifold to be totally geodesic has also been explored.

Bakry–Émery curvature and model spaces in sub-Riemannian geometry

We prove comparison theorems for the sub-Riemannian distortion coefficients appearing in interpolation inequalities. These results, which are equivalent to a sub-Laplacian comparison theorem for the sub-Riemannian distance, are obtained by introducing a suitable notion of sub-Riemannian Bakry-\'Emery curvature. The model spaces for comparison are variational problems coming from optimal control theory. As an application we establish the sharp measure contraction property for 3-Sasakian manifolds satisfying a suitable curvature bound.

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.

Locally homogeneous aspherical Sasaki manifolds

Let G/H be a contractible homogeneous Sasaki manifold. A compact locally homogeneous aspherical Sasaki manifold Γ\\G/H is by definition a quotient of G/H by a discrete uniform subgroup Γ≤G. We show that a compact locally homogeneous aspherical Sasaki manifold is always quasi-regular, that is, Γ\\G/H is an S1-Seifert bundle over a locally homogeneous aspherical Kähler orbifold. We discuss the structure of the isometry group Isom(G/H) for a Sasaki metric of G/H in relation with the pseudo-Hermitian group Psh(G/H) for the Sasaki structure of G/H. We show that a Sasaki Lie group G, when Γ\\G is a compact locally homogeneous aspherical Sasaki manifold, is either the universal covering group of SL(2,R) or a modification of a Heisenberg nilpotent Lie group with its natural Sasaki structure. We also show that any compact regular aspherical Sasaki manifold with solvable fundamental group is finitely covered by a Heisenberg manifold and its Sasaki structure may be deformed to a locally homogeneous one. In addition, we classify all aspherical Sasaki homogeneous spaces for semisimple Lie groups.