Sasaki Metric Research Articles

Sasakian geometry on sphere bundles

The purpose of this paper is to study the Sasakian geometry on odd dimensional sphere bundles over a smooth projective algebraic variety N with the ultimate, but probably unachievable goal of understanding the existence and non-existence of extremal and constant scalar curvature Sasaki metrics. We apply the fiber join construction of Yamazaki [48] for K-contact manifolds to the Sasaki case. This construction depends on the choice of d+1 integral Kähler classes [ωj] on N that are not necessarily colinear in the Kähler cone. We show that the colinear case is equivalent to a subclass of a different join construction originally described in [7,11], applied to the spherical case by the authors in [19,20] when d=1, and known as cone decomposable [13]. The non-colinear case gives rise to infinite families of new inequivalent cone indecomposable Sasaki contact CR structures on certain sphere bundles. We prove that the Sasaki cone for some of these structures contains an open set of extremal Sasaki metrics and, for certain specialized cases, the regular ray within this cone is shown to have constant scalar curvature. We also compute the cohomology groups of all such sphere bundles over a product of Riemann surfaces.

Berger Type Deformed Sasaki Metric and Harmonicity on the Cotangent Bundle

In this paper, we introduce the Berger type deformed Sasaki metric on the cotangent bundle $T^{\ast}M$ over an anti-paraKähler manifold $(M, \varphi, g)$. We establish a necessary and sufficient conditions under which a covector field is harmonic with respect to the Berger type deformed Sasaki metric. We also construct some examples of harmonic vector fields. we also study the harmonicity of a map between a Riemannian manifold and a cotangent bundle of another Riemannian manifold and vice versa.

K-Contact and Sasakian metrics as Ricci almost solitons

We show that the scalar curvature of a [Formula: see text]-contact Ricci soliton is constant and satisfies sharp bounds. Next we show that the scalar curvature of a ([Formula: see text])-dimensional [Formula: see text]-contact Ricci almost soliton is equal to [Formula: see text] plus the divergence of a global vector field. Finally, we show that, if a complete connected Sasakian or [Formula: see text]-Einstein [Formula: see text]-contact manifold of dimension [Formula: see text] is a proper Ricci almost soliton, then it is isometric to a unit sphere.

Twisted Sasakian Metric on the Tangent Bundle and Harmonicity

In the present paper, we introduce a new class of natural metrics on the tangent bundle $TM$ of the Riemannian manifold $(M,g)$ denoted by $G^{f,h}$ which is named a twisted Sasakian metric. A necessary and sufficient conditions under which a vector field is harmonic with respect to the twisted Sasakian metric are established. Some examples of harmonic vector fields are presented as well.

Geodesics of fiberwise cigar soliton deformation of the Sasaki metric

Ricci solitons arose in proof the Poincare conjecture by R. Hamilton and G. Perelman. The first example of a noncompact steady Ricci soliton on a plane was found by R. Hamilton. This two-dimensional manifold is conformally equivalent to the plane and it is called by R. Hamilton's cigar soliton. The cigar soliton metric can be considered as a fiber-wise conformal deformation of the Euclidean metric on a fiber of the tangent bundle. In the paper we propose a deformation of the classical Sasaki metric on the tangent bundle of an n-dimensional Riemannian manifold that induces the cigar soliton type metric on the fibers. The purpose of the research is to study geodesics of the cigar soliton deformation of the Sasaki metric on the tangent bundle of the Riemannian manifold with focus on the locally symmetric/constant curvature base manifold.

Almost Hermitian structures defining harmonic maps of the unit tangent bundle

A compatible almost complex structure on a Riemannian manifold can be considered as a smooth map J from the manifold into its twistor space endowed with a natural Riemannian metric induced by the metric of the base manifold or as a self-map J of the unit tangent bundle endowed with the Sasaki metric. The conditions under which the map J is harmonic have been found in Davidov et al. (2018) in the four-dimensional case. In the present paper, we discuss the problem of when the map J is harmonic and show that if J is harmonic, so is J, but not vice versa. Also, we obtain geometric conditions under which J is a harmonic map in the case when the Riemannian manifold is of dimension four.

The Geometry of the Sasaki Metric on the Sphere Bundles of Euclidean Atiyah Vector Bundles

Let $$(M,\langle \;,\;\rangle _{TM})$$ be a Riemannian manifold. It is well known that the Sasaki metric on TM is very rigid, but it has nice properties when restricted to $$T^{(r)}M=\{u\in TM,|u|=r \}$$ . In this paper, we consider a general situation where we replace TM by a vector bundle $$E\longrightarrow M$$ endowed with a Euclidean product $$\langle \;,\;\rangle _E$$ and a connection $$\nabla ^E$$ which preserves $$\langle \;,\;\rangle _E$$ . We define the Sasaki metric on E and we consider its restriction h to $$E^{(r)}=\{a\in E,\langle a,a\rangle _E=r^2 \}$$ . We study the Riemannian geometry of $$(E^{(r)},h)$$ generalizing many results first obtained on $$T^{(r)}M$$ and establishing new ones. We apply the results obtained in this general setting to the class of Euclidean Atiyah vector bundles introduced by the authors in Boucetta and Essoufi J Geom Phys 140:161–177, 2019). Finally, we prove that any unimodular three dimensional Lie group G carries a left invariant Riemannian metric, such that $$(T^{(1)}G,h)$$ has a positive scalar curvature.

The CR geometry of weighted extremal Kähler and Sasaki metrics

We establish an equivalence between conformally Einstein–Maxwell Kahler 4-manifolds recently studied in Apostolov et al. (J Reine Angew Math 721:109–147, 2016), Apostolov and Maschler (J Eur Math Soc 21:1319–1360, 2019), Futaki and Ono (J Math Soc Jpn 70:1493–1521, 2018), Koca et al. (Ann Glob Anal Geom 50, 29–46, 2016), LeBrun (Einstein–Maxwell equations, extremal Kahler metrics, and Seiberg–Witten theory in “The Many Facets of Geometry: A Tribute to Nigel Hitchin”, Oxford University Press, Oxford, pp 17–33, 2009), LeBrun (J Geom Phys 91:–171, 2015) and LeBrun (Commun Math Phys 344:621–653, 2016) and extremal Kahler 4-manifolds in the sense of Calabi (Extremal Kahler metrics, seminar on differential geometry, annals of mathematics studies, vol 102, pp 259–290, Princeton University Press, Princeton, 1982) with nowhere vanishing scalar curvature. The corresponding pairs of Kahler metrics arise as transversal Kahler structures of Sasaki metrics compatible with the same CR structure and having commuting Sasaki–Reeb vector fields. This correspondence extends to higher dimensions using the notion of a weighted extremal Kahler metric (Apostolov et al. in Levi–Kahler reduction of CR structures, product of spheres, and toric geometry, arXiv:1708.05253 ; Apostolov et al. in Weighted extremal Kahler metrics and the Einstein–Maxwell geometry of projective bundles, arXiv:1808.02813 ; Lahdili in J Geom Anal 29:542–568, 2019; Lahdili in Int Math Res Not, arXiv:1710.00235 ; Lahdili in Proc Lond Math Soc 119:1065–1114, 2019) illuminating and uniting several explicit constructions in Kahler and Sasaki geometry. It also leads to new existence and non-existence results for extremal Sasaki metrics, suggesting a link between the notions of relative weighted K-stability of a polarized variety introduced in Apostolov et al. (Weighted extremal Kahler metrics and the Einstein–Maxwell geometry of projective bundles, arXiv:1808.02813 ) and Lahdili (Proc Lond Math Soc 119:1065–1114, 2019), and relative K-stability of the Kahler cone corresponding to a Sasaki polarization, studied in Boyer and van Coevering (Math Res Lett 25:1–19, 2018) and Collins and Szekelyhidi (J Differ Geom 109:81–109, 2018).

Some Problems Concerning with Sasaki Metric on the Second-Order Tangent Bundles

In this paper, we consider a second-order tangent bundle equipped with
 Sasaki metric over a Riemannian manifold. All forms of curvature tensor
 fields are computed. We obtained the relation between the scalar curvature
 of the base manifold and the scalar curvature of the second-order tangent
 bundle and presented some geometric results concerning with kinds of
 curvature tensor fields. Also, we search the weakly symmetry property of the
 second-order tangent bundle. Finally, we end our paper with statistical
 structures on the second-order tangent bundle.

Generalized quasi-statistical structures

Given a non-degenerate $(0,2)$-tensor field $h$ on a smooth manifold $M$, we consider a natural generalized complex and a generalized product structure on the generalized tangent bundle $TM\oplus T^*M$ of $M$ and we show that they are $\nabla$-integrable, for $\nabla$ an affine connection on $M$, if and only if $(M,h,\nabla)$ is a quasi-statistical manifold. We introduce the notion of generalized quasi-statistical structure and we prove that any quasi-statistical structure on $M$ induces generalized quasi-statistical structures on $TM\oplus T^*M$. In this context, dual connections are considered and some of their properties are established. The results are described in terms of Patterson-Walker and Sasaki metrics on $T^*M$, horizontal lift and Sasaki metrics on $TM$ and, when the connection $\nabla$ is flat, we define the prolongations of the quasi-statistical structures on the manifolds to their cotangent and tangent bundles via Generalized Geometry. Moreover, Norden and Para-Norden structures are defined on $T^*M$ and $TM$.

Gradient Weyl-Ricci Soliton

The classical notion of gradient Ricci soliton is extended here to the gradient Weyl-Ricci soliton. A Weyl structureofthebasemanifold M is lifted to its tangent bundle TM, by using the Sasaki metric. We give some necessary and sufficient conditions such that the Weyl structure on TM to be a gradient Weyl-Ricci soliton.

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.

Ricci solitons on three dimensional generalized sasakian space forms with quasi Sasakian metric

The object of the present paper is to study Ricci solitons on three dimensional generalized Sasakian space forms with quasi Sasakian metric. We have also studied gradient Ricci solitons on such manifolds. Examples have been given.

English

We find all the left-invariant harmonic unit vector fields on the oscillator groups. Besides, we determine the associated harmonic maps from the oscillator group into its unit tangent bundle equipped with the associated Sasaki metric. Moreover, we investigate the stability and instability of harmonic unit vector fields on compact quotients of four dimensional oscillator group G1(1).

The geometry of generalized Cheeger–Gromoll metrics on the total space of transitive Euclidean Lie algebroids

Natural metrics (Sasaki metric, Cheeger–Gromoll metric, Kaluza–Klein metrics etc. ) on the tangent bundle of a Riemannian manifold is a central topic in Riemannian geometry. Generalized Cheeger–Gromoll metrics is a family hp,q of natural metrics on the tangent bundle depending on two parameters with p∈R and q≥0. This family possesses interesting geometric properties. If p=q=0 we recover the Sasaki metric and when p=q=1 we recover the classical Cheeger–Gromoll metric. A transitive Euclidean Lie algebroid is a transitive Lie algebroid with a Euclidean product on its total space. In this paper, we show that the analogous of natural metrics 11Natural metrics on the tangent bundles come from a classification of natural transformation of Riemannian metrics on manifolds to metrics on tangent bundles. We use the some terminology in this paper to designate metrics obtained on the total space of a Euclidean Lie algebroid by imitating the expression of g-natural metrics on the tangent bundle.can be built in a natural way on the total space of transitive Euclidean Lie algebroids. Then we study the properties of generalized Cheeger–Gromoll metrics on this new context. We show a rigidity result of these metrics which generalizes some rigidity results known in the case of the tangent bundle. We show also that considering natural metrics on the total space of transitive Euclidean Lie algebroids opens new interesting horizons. For instance, Atiyah Lie algebroids constitute an important class of transitive Lie algebroids and we will show that natural metrics on the total space of Atiyah Euclidean Lie algebroids have interesting properties. In particular, if M is a Riemannian manifold of dimension n, then the Atiyah Lie algebroid associated to the O(n)-principal bundle of orthonormal frames over M possesses a family depending on a parameter k>0 of transitive Euclidean Lie algebroids structures say AO(M,k). When M is a space form of constant curvature c, we show that there exist two constants Cn<0 and K(n,c)>0 such that (AO(M,k),h1,1) is a Riemannian manifold with positive scalar curvature if and only if c>Cn and 0<k≤K(n,c).

Statistical structures on the tangent bundle of a statistical manifold with Sasaki metric

The first part of the paper is devoted to the classification of the statistical structures which live on the tangent bundle of a statistical manifold endowed with a Sasaki metric. Further, considering a Kahler structure on the base statistical manifold, we introduce a family of almost complex structures on the tangent bundle equipped with the Sasaki metric, and find equivalent conditions for which this family induces a Kahler structure. Finally, we derive equivalent conditions for existence of holomorphic structures on the tangent bundle equipped with the Sasaki metric in the presence of a statistical structure. Several illustrative examples are provided, as well.

Geometry of Twisted Sasaki Metric

Let $(M,g)$ be a $n$-dimensional smooth Riemannian manifold. In the present paper, we introduce a new class of natural metrics denoted by $G^{f,h}$ and called twisted Sasaki matric on the tangent bundle $TM$. We studied the geometry of $(TM,G^{f,h})$ by giving a relationships of the curvatures, Einstein structure, scalar and sectional curvatures between $(TM,G^{f,h})$ and $(M,g)$.

Contact Structures of Sasaki Type and Their Associated Moduli

Abstract This article is based on a talk at the RIEMain in Contact conference in Cagliari, Italy in honor of the 78th birthday of David Blair one of the founders of modern Riemannian contact geometry. The present article is a survey of a special type of Riemannian contact structure known as Sasakian geometry. An ultimate goal of this survey is to understand the moduli of classes of Sasakian structures as well as the moduli of extremal and constant scalar curvature Sasaki metrics, and in particular the moduli of Sasaki-Einstein metrics.

Reducibility in Sasakian geometry

The purpose of this paper is to study reducibility properties in Sasakian geometry. First we give the Sasaki version of the de Rham decomposition theorem; however, we need a mild technical assumption on the Sasaki automorphism group which includes the toric case. Next we introduce the concept of cone reducible and consider S 3 S^3 bundles over a smooth projective algebraic variety where we give a classification result concerning contact structures admitting the action of a 2-torus of Reeb type. In particular, we can classify all such Sasakian structures up to contact isotopy on S 3 S^3 bundles over a Riemann surface of genus greater than zero. Finally, we show that in the toric case an extremal Sasaki metric on a Sasaki join always splits.

∗-Ricci Soliton within the frame-work of Sasakian and (κ,μ)-contact manifold

We prove that if a Sasakian metric is a ∗-Ricci Soliton, then it is either positive Sasakian, or null-Sasakian. Next, we prove that if a complete Sasakian metric is an almost gradient ∗-Ricci Soliton, then it is positive-Sasakian and isometric to a unit sphere [Formula: see text]. Finally, we classify nontrivial ∗-Ricci Solitons on non-Sasakian [Formula: see text]-contact manifolds.