Charles P. Boyer, Christina W. Tønnesen-Friedman
We prove the existence of extremal Sasakian structures occurring on a countably infinite number of distinct contact structures on $T^2\times S^3$ and certain related manifolds. These structures occur in bouquets and exhaust the Sasaki cones in all except one case in which there are no extremal metrics.
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We study the Sasaki cone of a CR structure of Sasaki type on a given closed manifold. We introduce an energy functional over the cone, and use its critical points to single out the strongly extremal Reeb vectors fields. Should one such vector field be a member of the extremal set, the scalar curvature of a Sasaki extremal metric representing it would have the smallest $L^2$-norm among all Sasakian metrics of fixed volume that can represent vector fields in the cone. We use links of isolated hypersurface singularities to produce examples of manifolds of Sasaki type, many of these in dimension f…
In this note I study the Sasakian geometry associated to the standard CR structure on the Heisenberg group, and prove that the Sasaki cone coincides with the set of extremal Sasakian structures. Moreover, the scalar curvature of these extremal metrics is constant if and only if the metric has $Φ$-sectional curvature $-3.$ I also briefly discuss some relations with the well-know sub-Riemannian geometry of the Heisenberg group as well as the standard Sasakian structure induced on compact quotients.
In this note I prove the existence of extremal Sasakian metrics on S^3-bundles over S^2. These occur in a collection of open cones that I call a bouquet.
We show that the contact reduction can be specialized to Sasakian manifolds. We link this Sasakian reduction to Kähler reduction by considering the Kähler cone over a Sasakian manifold. We present examples of Sasakian manifolds obtained by $S^1$ reduction of standard Sasakian spheres.
The purpose of this paper is to study *-Ricci tensor on Sasakian manifold. Here, ϕ-confomally flat and confomally flat *-η-Einstein Sasakian manifold are studied. Next, we consider *-Ricci symmetric conditon on Sasakian manifold. Finally, we study a special type of metric called *-Ricci soliton on Sasakian manifold.
In the present paper first, we define the conformal Sasakian manifolds and then we study geometry of invariant, anti-invariant and CR-submanifolds of conformal Sasakian manifolds.
In this study, we introduce indefinite sasakian statistical manifolds and lightlike hypersurfaces of an indefinite sasakian statistical manifold. Some relations among induced geometrical objects with respect to dual connections in a lightlike hypersurface of an indefinite sasakian manifold are obtained. Some examples related to these concepts are also presented. Finally, we prove that an invariant lightlike submanifold of indefinite sasakian statistical manifold is an indefinite sasakian statistical manifold.
The orthogonal decomposition of the Webster curvature provides us a way to characterize some canonical metrics on a pseudo-Hermitian manifold. We derive some subelliptic differential inequalities from the Weitzenböck formulas for the traceless pseudo-Hermitian Ricci tensor and the Chern-Moser tensor of Sasakian manifolds with constant pseudo-Hermitian scalar curvature and Sasakian pseudo-Einstein manifolds respectively. By means of either subelliptic estimates or maximum principle, some rigidity theorems are established to characterize Sasakian pseudo-Einstein manifolds among Sasakian manifold…
In this short note we present some remarks concerning anti-quasi-Sasakian manifolds. Some proofs of their basic properties are simplified. We also discuss some canonical invariant distributions which exist on every anti-quasi-Sasakian manifold.
We provide a general method to construct examples of quasi-Sasakian 3-structures on a (4n+3)-dimensional manifold. Moreover, among this class, we give the first explicit example of a compact 3-quasi-Sasakian manifold which is not the global product of a 3-Sasakian manifold and a hyper-Kähler manifold.
We investigate the analog of holomorphic vector bundles in the context of Sasakian manifolds.
We consider some natural infinitesimal Einstein deformations on Sasakian and 3-Sasakian manifolds. Some of these are infinitesimal deformations of Killing spinors and further some integrate to actual Killing spinor deformations. In particular, on 3-Sasakian 7 manifolds these yield infinitesimal Einstein deformations preserving 2, 1, or none of the 3 independent Killing spinors. Toric 3-Sasakian manifolds provide non-trivial examples with integrable deformation preserving precisely 2 Killing spinors. Thus the dimension of Killing spinors is not preserved under Einstein deformations but is only …
The present paper is devoted to quasi-Para-Sasakian manifolds. Basic properties of such manifolds are obtained and general curvature identities are investigated. Next it is proved that if $M$ is quasi-Para-Sasakian manifold of constant curvature $K$. Then $K$ $\leq 0$ and $(i)~$if $K=0$, the manifold is paracosymplectic, $(ii)$ if $K<0$, the quasi-para-Sasakian structure of $M$ is obtained by a homothetic deformation of a para-Sasakian structure.
In this paper, we obtain some sufficient conditions for a 3-dimensional compact trans-Sasakian manifold of type $(α,β)$ to be homothetic to a Sasakian manifold. A characterization of a 3-dimensional cosymplectic manifold is also obtained.
In this paper, we introduce the trans-para-Sasakian manifolds and we study their geometry. These manifolds are an analogue of the trans-Sasakian manifolds in the Riemannian geometry. We shall investigate many curvature properties of these manifolds and we shall give many conditions under which the manifolds are either $η-$Einstein or Einstein manifolds.
In this paper we study the 3-dimensional $(\varepsilon) $-para Sasakian manifolds. We obtain an necessary and sufficient condition for an $(\varepsilon ) $-para Sasakian 3 -manifold to be an indefinite space form. We show that a Ricci-semi-symmetric $(\varepsilon) $-para Sasakian 3 -manifold is an indefinite space form. We investigate the necessary and sufficient condition for an $(\varepsilon) $-para Sasakian 3 -manifold to be locally $φ$-symmetric. It is proved that in an $ (\varepsilon) $-para Sasakian 3-manifold with $η$ -parallel Ricci tensor the scalar curvature is constant. It is also s…
This paper is devoted to the study of curvature and torsion of almost contact curves in trans-Sasakian 3-Manifolds. The conditions for the frenet curves to be almost contact curves in trans-Sasakian 3-manifolds have been obtained.
We prove that every nearly Sasakian manifold of dimension greater than five is Sasakian. This provides a new criterion for an almost contact metric manifold to be Sasakian. Moreover, we classify nearly cosymplectic manifolds of dimension greater than five.
We extend a result of Patodi for closed Riemannian manifolds to the context of closed contact manifolds by showing the condition that a manifold is an $η$-Einstein Sasakian manifold is spectrally determined. We also prove that the condition that a Sasakian space form has constant $ϕ$-sectional curvature $c$ is spectrally determined.
We show that a Sasakian metric which also satisfies the gradient Ricci soliton equation is necessarily Einstein.
We find some curvature properties of 3-quasi-Sasakian manifolds which are similar to some well-known identities holding in the Sasakian case. As an application, we prove that any 3-quasi-Sasakian manifold of constant horizontal sectional curvature is necessarily either 3-α-Sasakian or 3-cosymplectic.
The object of the present paper is to study 3-dimensional conformally flat quasi-Para-Sasakian manifolds. First, the necessary and sufficient conditions are provided for 3-dimensional quasi-Para-Sasakian manifolds to be conformally flat. Next, a characterization of 3-dimensional conformally flat quasi-Para-Sasakian manifold with \b{eta}=const. is given.
In this paper we study *-Conformal η-Ricci soliton on Sasakian manifolds. Here, we discuss some curvature properties on Sasakian manifold admitting *-Conformal η-Ricci soliton. We obtain some significant results on *-Conformal η-Ricci soliton in Sasakian manifolds satisfying R(ξ,X).S = 0, S(ξ,X).R = 0, {\overline}P(ξ,X).S = 0, where {\overline}P is Pseudo-projective curvature tensor.The conditions for *-Conformal η-Ricci soliton on Φ-conharmonically flat and Φ-projectively flat Sasakian manifolds have been obtained in this article. Lastly we give an example of 5-dimensional Sasakian manifolds …
In this paper, we study the invariant and noninvariant hypersurfaces of (1,1,1) almost contact manifolds, Lorentzian almost paracontact manifolds and Lorentzian para-Sasakian manifolds, respectively. We show that a noninvariant hypersurface of an (1,1,1) almost contact manifold admits an almost product structure. We investigate hypersurfaces of affinely cosymplectic and normal (1,1,1) almost contact manifolds. It is proved that a noninvariant hypersurface of a Lorentzian almost paracontact manifold is an almost product metric manifold. Some necessary and sufficient conditions have been given f…
The object of this paper is study $(ε)$-para-Sasakian 3-manifolds satisfying certain conditions on the $\mathcal{Z}$ tensor. We characterize, $\mathcal{Z}$-symmetric; $\mathcal{Z}$-semisymmetric; $\mathcal{Z}$-pseudosymmetric; and projectively $\mathcal{Z}$-semisymmetric conditions on an $(ε)$-para-Sasakian 3-manifold.
The object of the present paper is to study some properties of 3-dimensional trans-Sasakian manifold whose metric is η-Yamabe soliton. We have studied here some certain curvature conditions of 3-dimensional trans-Sasakian manifold admitting η-Yamabe soliton. Lastly we construct a 3-dimensional trans-Sasakian manifold satisfying η-Yamabe soliton.
We provide a new, self-contained and more conceptual proof of the result that an almost contact metric manifold of dimension greater than 5 is Sasakian if and only if it is nearly Sasakian.
In this paper we study a special type of metric called *-Ricci soliton on para-Sasakian manifold. We prove that if the para-Sasakian metric is a *-Ricci soliton on a manifold M, then M is either D-homothetic to an Einstein manifold, or the Ricci tensor of M with respect to the canonical paracontact connection vanishes.
We introduce slant Riemannian submersions from Sasakian manifolds onto Riemannian manifolds. We survey main results of slant Riemannian submersions defined on Sasakian manifolds. We also give an example of such slant submersions.
The purpose of the present paper is to study the globally and locally $φ$-${\cal T}$-symmetric $\left( \varepsilon \right) $-para Sasakian manifold in dimension $3$. The globally $φ$-$ {\cal T}$-symmetric $3$-dimensional $\left( \varepsilon \right) $-para Sasakian manifold is either Einstein manifold or has a constant scalar curvature. The necessary and sufficient condition for Einstein manifold to be globally $φ$-${\cal T}$ -symmetric is given. A $3$-dimensional $% \left( \varepsilon \right) $ -para Sasakian manifold is locally $φ$-$ {\cal T}$-symmetric if and only if the scalar curvature $r$…
In this paper we study the Sasakian geometry on S^3-bundles over a Riemann surface of genus g>0 with emphasis on extremal Sasaki metrics. We prove the existence of a countably infinite number of inequivalent contact structures on the total space of such bundles that admit 2-dimensional Sasaki cones each with a Sasaki metric of constant scalar curvature (CSC). This CSC Sasaki metric is most often irregular. We further study the extremal subset in the Sasaki cone showing that if 0<g<5 it exhausts the entire cone. Examples are given where exhaustion fails.
In this article, we introduce some metallic structures on the tangent bundle of a P-Sasakian manifold by complete lift, horizontal lift and vertical lift of a P-Sasakian structure $(ϕ, η,ξ)$ on tangent bundle. Then we investigate the integrability and parallelity of these metallic structures.
The object of the present paper is to study locally $ϕ$-symmetric LP-Sasakian manifolds admitting semi-symmetric metric connection and obtain a necessary and sufficient condition for a locally $ϕ$-symmetric LP-Sasakian manifold with respect to semi-symmetric metric connection to be locally $ϕ$-symmetric LP-Sasakian manifold with respect to Levi-Civita connection.
We present a non existence result of complete, Einstein hypersurfaces tangent to the Reeb vector field of a regular Sasakian manifold which fibers onto a complex Stein manifold.
Sasakian manifolds provide explicit formulae of some Jacobi operators which describe the biharmonic equation of curves in Riemannian manifolds. In this paper we characterize non-geodesic biharmonic curves in Sasakian manifolds which are either tangent or normal to the Reeb vector field. In the three-dimensional case, we prove that such curves are some helixes whose geodesic curvature and geodesic torsion satisfy a given relation.
3-quasi-Sasakian manifolds were studied systematically by the authors in a recent paper as a suitable setting unifying 3-Sasakian and 3-cosymplectic geometries. This paper throws new light on their geometric structure which reveals to be generally richer compared to the 3-Sasakian subclass. In fact, it turns out that they are multiply foliated by four distinct fundamental foliations. The study of the transversal geometries with respect to these foliations allows us to link the 3-quasi-Sasakian manifolds to the more famous hyper-Kaehler and quaternionic-Kaehler geometries. Furthermore, we stron…
Nilpotent aspherical Sasakian manifolds
本文证明了具有幂零基本群的紧致非球面Sasakian流形必微分同胚于Heisenberg幂零流形，从而给出该类流形的完整分类。
We prove a Bishop volume comparison theorem and a Laplacian comparison theorem for a natural sub-Riemannian structure defined on Sasakian manifolds. This generalizes the earlier work for the three dimensional case.
In this paper we provide a second variation formula for L-minimal Lagrangian submanifolds in a pseudo-Sasakian manifold. We apply it to the case of Lorentzian-Sasakian manifolds and relate the L-stability of L-minimal Legendrians in a Sasakian M to their L-stability in an associated Lorentzian-Sasakian structure on M.
In this paper, we derive a sub-gradient estimate for pseudoharmonic maps from noncompact complete Sasakian manifolds which satisfy CR sub-Laplace comparison property, to simply-connected Riemannian manifolds with nonpositive sectional curvature. As its application, we obtain some Liouville theorems for pseudoharmonic maps. In the Appendix, we modify the method and apply it to harmonic maps from noncompact complete Sasakian manifolds.
We prove that any simply connected compact 3-Sasakian manifold, of dimension seven, is formal if and only if its second Betti number is $b_2<2$. In the opposite, we show an example of a 7-dimensional Sasaki-Einstein manifold, with second Betti number $b_2\geq 2$, which is formal. Therefore, such an example does not admit any 3-Sasakian structure. Examples of 7-dimensional simply connected compact formal Sasakian manifolds, with $b_2\geq 2$, are also given.
Hitchin sections on $3$-dimensional Sasakian manifolds
利用非阿贝尔Hodge对应，在三维紧致Sasakian流形上构造秩2的Hitchin截面，通过典范形变建立基本二次微分空间与特定Sasakian结构等价类空间的微分同胚。
In this article we study a class of normalθcomplexθcontactθmetricθmanifold 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.
Generalized $η-$Ricci solitons on LP-Sasakian manifolds admitting the general connection
本文研究了在允许一般连接的LP-Sasakian流形上广义η-Ricci溶剂的性质，并通过构造例子证明了其存在性。
In this note, We proved that a compact Sasakian manifold $( M, ξ, η, Φ , g )$ with negative transverse holomorphic sectional curvature must have has a Sasakian structure $( ξ, η , Φ , g )$ with negative transverse Ricci curvature. Similarly, a compact Sasakian manifold with non-positive transverse holomorphic sectional curvature, then the negative first basic Chern class is transverse nef and we have the Miyaoka-Yau type inequality. When transverse holomorphic sectional curvature is quasi-negative, we obtain a Chern number inequality.
We build a variational theory of geodesics of the Tanaka-Webster connection on a strictly pseudoconvex CR manifold.
In this paper we provide a local construction of a Sasakian manifold given a Kähler manifold. Obatined in this way manifold we call Sasakian lift of Kähler base. Almost contact metric structure is determined by the operation of the lift of vector fields - idea similar to lifts in Ehresmann connections. We show that Sasakian lift inherits geometry very close to its Kähler base. In some sense geometry of the lift is in analogy with geometry of hypersurface in Kähler manifold. There are obtained structure equations between corresponding Levi-Civita connections, curvatures and Ricci tensors of the…
Warped product CR-submanifolds in Kaehlerian manifolds were intensively studied only since 2001 after the impulse given by B.Y. Chen. Immediately after, another line of research, similar to that concerning Sasakian geometry as the odd dimensional version of Kaehlerian geometry, was developed, namely warped product contact CR-submanifolds in Sasakian manifolds. In this note we proved that there exists no proper doubly warped product contact CR-submanifolds in trans-Sasakian manifolds.
We provide a new, self-contained proof of the classification of homogeneous 3-Sasakian manifolds, which was originally obtained by Boyer, Galicki and Mann. In doing so, we construct an explicit one-to-one correspondence between simply connected homogeneous 3-Sasakian manifolds and simple complex Lie algebras via the theory of root systems. We also discuss why the real projective spaces are the only non-simply connected homogeneous 3-Sasakian manifolds and derive the famous classification of homogeneous positive quaternionic Kähler manifolds due to Wolf and Alekseevskii from our results.
We introduce anti-invariant Riemannian submersions from Sasakian manifolds onto Riemannian manifolds. We survey main results of anti-invariant Riemannian submersions defined on Sasakian manifolds. We investigate necessary and sufficient condition for an anti-invariant Riemannian submersion to be totally geodesic and harmonic. We give examples of anti-invariant submersions such that characteristic vector field ξ is vertical or horizontal.
Immersions of Sasaki-Ricci solitons into homogeneous Sasakian manifolds
该研究探讨Sasaki-Ricci孤立子到齐次Sasakian流形纤维积的局部浸入，证明一类纤维积诱导的孤立子为η-Einstein，浸入空间形式时结果更强，并给出无局部全纯等距映射的Kähler-Ricci孤立子例子。
The Newman-Penrose-Perjes formalism is applied to Sasakian 3-manifolds and the local form of the metric and contact structure is presented. The local moduli space can be parameterised by a single function of two variables and it is shown that, given any smooth function of two variables, there exists locally a Sasakian structure with scalar curvature equal to this function. The case where the scalar curvature is constant ($η$-Einstein Sasakian metrics) is completely solved locally. The resulting Sasakian manifolds include $S^3$, $Nil$ and $\tilde{SL_2R}$, as well as the Berger spheres. It is al…
We propose a new method to construct degenerate $3$-$(α,δ)$-Sasakian manifolds as fiber products of Boothby-Wang bundles over hyperkähler manifolds. Subsequently, we study homogeneous degenerate $3$-$(α, δ)$-Sasakian manifolds and prove that no non-trivial compact examples exist as well as that there is exactly one family of nilpotent Lie groups with this geometry, the quaternionic Heisenberg groups.
We construct a compact simply-connected 7-dimensional manifold admitting a K-contact structure but not a Sasakian structure. We also study rational homotopy properties of such manifolds, proving in particular that a simply-connected 7-dimensional Sasakian manifold has vanishing cup-product on the second cohomology and that it is formal if and only if all its triple Massey products vanish.
Clairaut anti-invariant Riemannian maps to trans-Sasakian manifolds
该研究引入到trans-Sasakian流形的Clairaut反不变黎曼映射，推导其充要条件，讨论相关分布可积性与映射调和性，并构造非平凡实例验证。
Immersions into Sasakian space forms
本文研究Sasakian流形到Sasakian空间形式的浸入，在证明Calabi刚性结果后，刻画了可局部浸入非椭圆空间形式及在紧致与非紧致情形下可嵌入标准球面的齐次Sasakian流形。
On the curvature symmetries in lightlike hypersurfaces of an indefinite Sasakian manifold
本文研究不定 Sasakian 流形中相切于结构向量场的类光超曲面，证明其不具备局部对称、半对称或半平行等曲率对称性。
In this paper we have studied the properties of Quasi umbilical hypersurface $M$ of a Sasakian manifold $\tilde M$ with $(ϕ, g, u, v, λ)-$structure and established the relation for $M$ to be cylindrical.
The purpose of this paper is to study the harmonicity of maps to or from para-Sasakian manifolds. We derive the condition for the tension field of paraholomorphic map between almost para-Hermitian manifold and para-Sasakian manifold. The necessary and sufficient condition for a paraholomorphic map between para-Sasakian manifolds to be parapluriharmonic are shown and a non-trivial example is presented for its illustrations.
We define pseudo-Hermitian magnetic curves in Sasakian manifolds endowed with the Tanaka-Webster connection. After we give a complete classification theorem, we construct parametrizations of pseudo-Hermitian magnetic curves in $\mathbb{R}^{2n+1}(-3)$.
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