In this paper, we show that the existence of Sasakian-Einstein metrics is closely related to the properness of corresponding energy functionals. Under the condition that admitting no nontrivial Hamiltonian holomorphic vector field, we prove that the existence of Sasakian-Einstein metric implies a Moser-Trudinger type inequality. At the end of this paper, we also obtain a Miyaoka-Yau type inequality in Sasakian geometry.
The aim of this paper is to study Sasakian immersions of (non-compact) complete regular Sasakian manifolds into the Heisenberg group and into $ \mathbb{B}^N\times \mathbb{R}$ equipped with their standard Sasakian structures. We obtain a complete classification of such manifolds in the $η$-Einstein case.
In this paper, we apply the method developed in [Ti97] and [TZ00] to proving the properness of log $F$-functional on any conic Kähler-Einstein manifolds. As an application, we give an alternative proof for the openness of the continuity method through conic Kähler-Einstein metrics.
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 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.
We show that $\scriptstyle{#9(S^2\times S^3)}$ admits an 8-dimensional complex family of inequivalent non-regular Sasakian-Einstein structures. These are the first known Einstein metrics on this 5-manifold. In particular, the bound $\scriptstyle{b_2(M)\leq8}$ which holds for any regular Sasakian-Einstein $\scriptstyle{M}$ does not apply to the non-regular case. We also discuss the failure of the Hitchin-Thorpe inequality in the case of 4-orbifolds and describe the orbifold version.
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 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…
The aim of this paper is to study Sasakian immersions of compact Sasakian manifolds into the odd-dimensional sphere equipped with the standard Sasakian structure. We obtain a complete classification of such manifolds in the Einstein and $η$-Einstein cases when the codimension of the immersion is $4$. Moreover, we exhibit infinite families of compact Sasakian $η$--Einstein manifolds which cannot admit a Sasakian immersion into any odd-dimensional sphere. Finally, we show that, after possibly performing a ${\mathcal {D}}$-homothetic deformation, a homogeneous Sasakian manifold can be Sasakian im…
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.
We carry on a systematic study of nearly Sasakian manifolds. We prove that any nearly Sasakian manifold admits two types of integrable distributions with totally geodesic leaves which are, respectively, Sasakian or $5$-dimensional nearly Sasakian manifolds. As a consequence, any nearly Sasakian manifold is a contact manifold. Focusing on the $5$-dimensional case, we prove that there exists a one-to-one correspondence between nearly Sasakian structures and a special class of nearly hypo $SU(2)$-structures. By deforming such a $SU(2)$-structure one obtains in fact a Sasaki-Einstein structure. Fu…
We discuss Sasakian-Einstein geometry under a quasi-regularity assumption. It is shown that the space of all quasi-regular Sasakian-Einstein orbifolds has a natural multiplication on it. Furthermore, necessary and sufficient conditions are given for the `product' of two Sasakian-Einstein manifolds to be a smooth Sasakian-Einstein manifold. Using spectral sequence arguments we work out the cohomology ring in many cases of interest. This type of geometry has recently become of interest in the physics of supersymmetric conformal field theories.
Let $S$ be a compact Sasakian manifold which does not admit non-trivial Hamiltonian holomorphic vector fields. If there exists an Einstein-Sasakian metric on $S$, then it is unique.
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 prove the following results: (i) A Sasakian metric as a non-trivial Ricci soliton is null $η$-Einstein, and expanding. Such a characterization permits to identify the Sasakian metric on the Heisenberg group $\mathcal{H}^{2n+1}$ as an explicit example of (non-trivial) Ricci soliton of such type. (ii) If an $η$-Einstein contact metric manifold $M$ has a vector field $V$ leaving the structure tensor and the scalar curvature invariant, then either $V$ is an infinitesimal automorphism, or $M$ is $D$-homothetically fixed $K$-contact.
A 3-Sasakian structure on a 7-manifold may be used to define two distinct Einstein metrics: the 3-Sasakian metric and the squashed Einstein metric. Both metrics are induced by nearly parallel $G_2$-structures which may also be expressed in terms of the 3-Sasakian structure. Just as Einstein metrics are critical points for the Ricci flow up to rescaling, nearly parallel $G_2$-structures provide natural critical points of the (rescaled) geometric flows of $G_2$-structures known as the Laplacian flow and Laplacian coflow. We study each of these flows in the 3-Sasakian setting and see that their b…
We construct stationary solutions to the Einstein-Maxwell-current system by using the Sasakian manifold for the three-dimensional space. Both the magnetic field and the electric current in the solution are specified by the contact form of the Sasakian manifold. The solutions contain an arbitrary function that describes inhomogeneity of the number density of the charged particles, and the function determines the curvature of the space.
In this paper, we initiate the study of conformal $η$-Ricci soliton and almost conformal $η$-Ricci soliton within the framework of para-Sasakian manifold. We prove that if para-Sasakian metric admits conformal $η$-Ricci soliton, then the manifold is $η$-Einstein and either the soliton vector field $V$ is Killing or it leaves $ϕ$ invariant. Here, we have shown the characteristics of the soliton vector field $V$ and scalar curvature when the manifold admitting conformal $η$-Ricci soliton and vector field is pointwise collinear with the characteristic vector field $ξ$. Next, we show that a para-S…
The present paper is to deliberate the class of $3$-dimensional trans-Sasakian manifolds which admits $η$-Einstein solitons. We have studied $η$-Einstein solitons on $3$-dimensional trans-Sasakian manifolds where the Ricci tensors are Codazzi type and cyclic parallel. We have also discussed some curvature conditions admitting $η$-Einstein solitons on $3$-dimensional trans-Sasakian manifolds and the vector field is torse-forming. We have also shown an example of $3$-dimensional trans-Sasakian manifold with respect to $η$-Einstein soliton to verify our results.
We study eta-Einstein geometry as a class of distinguished Riemannian metrics on contact metric manifolds. In particular, we use a previous solution of the Calabi problem for Sasakian geometry to prove the existence of eta-Einstein structures on many different compact manifolds, including exotic spheres. We also relate these results to the existence of Einstein-Weyl structures.
The theory of ambient spaces is useful to define CR invariant objects, such as CR invariant powers of the sub-Laplacian, the $P$-prime operators, and $Q$-prime curvature. However in general, it is difficult to write down these objects in terms of the Tanaka-Webster connection. In this paper, we give those explicit formulas for CR manifolds satisfying an Einstein condition, called Sasakian $η$-Einstein manifolds. As an application, we study properties of the first and the second variation of the total $Q$-prime curvature at Sasakian $η$-Einstein manifolds.
We show that #8(S^2 times S^3) admits two 8-dimensional complex families of inequivalent non-regular Sasakian-Einstein structures. These are the first known non-regular Sasakian-Einstein metrics on this 5-manifold.
Einstein like $(\varepsilon)$-para Sasakian manifolds are introduced. For an $(\varepsilon) $-para Sasakian manifold to be Einstein like, a necessary and sufficient condition in terms of its curvature tensor is obtained. The scalar curvature of an Einstein like $(\varepsilon) $-para Sasakian manifold is obtained and it is shown that the scalar curvature in this case must satisfy certain differential equation. A necessary and sufficient condition for an $(\varepsilon) $-almost paracontact metric hypersurface of an indefinite locally Riemannian product manifold to be $(\varepsilon) $-para Sasaki…
Let $(g,X)$ be a Sasaki-Ricci soliton on a Sasakian manifold $S$. We prove that if $(S,g)$ admits a local Sasakian immersion in a Sasakian space form $S(N,c)$ of constant $ϕ$-sectional curvature $c$, then $S$ is $η$-Einstein and its $η$-Einstein constants are rational. Moreover, if $c\leq -3$, $S$ is locally equivalent to the Sasakian space form $S(n,c)$ and its $η$-Einstein constants are determined by $c$. Further results are obtained in the compact setting, i.e. when $c>-3$, under additional hypotheses.
We prove that any non-Sasakian contact metric (κ,μ)-space admits a canonical η-Einstein Sasakian or η-Einstein paraSasakian metric. An explicit expression for the curvature tensor fields of those metrics is given and we find the values of κand μfor which such metrics are Sasaki-Einstein and paraSasaki-Einstein. Conversely, we prove that, under some natural assumptions, a K-contact or K-paracontact manifold foliated by two mutually orthogonal, totally geodesic Legendre foliations admits a contact metric (κ,μ)-structure. Furthermore, we apply the above results to the geometry of tangent sphere b…
We construct symplectic and Kähler ray reduced spaces and discuss their relation with the Marsden-Weinstein (point) reduction. This Kähler reduction is well defined even when the momentum value is not totally isotropic. The compatibility of the ray reduction with the cone construction and the Boothby-Wang fibration is presented. Using the compatibility with the cone construction we provide the exact description of ray quotients of cotangent bundles. Some applications of the ray reduction to the study of conformal Hamiltonian systems are described. We also give necessary and sufficient conditio…
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.
Let $X_0$ be an affine variety with only normal isolated singularity $p$ and $π: X\to X_0$ a smooth resolution of the singularity with trivial canonical line bundle $K_X$. If the complement of the affine variety $X_0\backslash\{p\}$ is the cone $C(S)=\Bbb R_{>0}\times S$ of an Einstein-Sasakian manifold $S$, we shall prove that the crepant resolution $X$ of $X_0$ admits a complete Ricci-flat Kähler metric in every Kähler class in $H^2(X)$. We apply the continuity method for solving the Monge-Ampère equation to obtain a relevant existence theorem and a uniqueness theorem of Ricci-flat conical K…
In this paper we deal with two classes of mixed metric 3-structures, namely the mixed 3-Sasakian structures and the mixed metric 3-contact structures. Firstly we study some properties of the curvature of mixed 3-Sasakian structures, proving that any manifold endowed with such a structure is Einstein. Then we prove the identity between the class of mixed 3-Sasakian structures and the class of mixed metric 3-contact structures.
We review our study of Sasakian geometry as an agent for proving the existence of Einstein metrics on odd dimensional manifolds. Particular emphasis is given to the Sasakian structures occuring on links of isolated hypersurface singularities.
On simply connected five manifolds Sasakian-Einstein metrics coincide with Riemannian metrics admitting real Killing spinors which are of great interest as models of near horizon geometry for three-brane solutions in superstring theory [KW]. We expand on the recent work of Demailly and Kollár [DK] and Johnson and Kollár [JK1] who give methods for constructing Kähler-Einstein metrics on log del Pezzo surfaces. By [BG1] circle V-bundles over log del Pezzo surfaces with Kähler-Einstein metrics have Sasakian-Einstein metrics on the total space of the bundle. Here these simply connected 5-manifolds…
Using the Sasakian join construction with homology 3-spheres, we give a countably infinite number of examples of Sasakian manifolds with perfect fundamental group in all odd dimensions greater than 1. These have extremal Sasaki metrics with constant scalar curvature. Moreover, we present further examples of both Sasaki-Einstein and Sasaki-$η$-Einstein metrics.
In this paper we demonstrate the existence of Sasakian-Einstein structures on certain 2-connected rational homology 7-spheres. These appear to be the first non-regular examples of Sasakian-Einstein metrics on simply connected rational homology spheres. We also briefly describe the rational homology 7-spheres that admit regular positive Sasakian structures.
We extend the link between Einstein Sasakian manifolds and Killing spinors to a class of $η$-Einstein Sasakian manifolds, both in Riemannian and Lorentzian settings, characterising them in terms of generalised Killing spinors. We propose a definition of supersymmetric M-theory backgrounds on such a geometry and find a new class of such backgrounds, extending previous work of Haupt, Lukas and Stelle.
Exact solutions to an Einstein-Maxwell(E-D-M) system with an electric current are simple models of global cosmic magnetic phenomena in the universe. In this paper, we consider an E-D-M system with two chiral spinors $ ψ^\pm$ coupled with a gauge field $ A$ and an electromagnetic field $F=dA$. We construct a family of exact solutions to the E-D-M system on four-dimensional static Sasakian spacetimes. The electromagnetic field in the solutions is a contact magnetic field, and the chiral spinors are induced from Sasakian quasi-Killing spinors. We find closed universe models whose weak energy cond…
This paper is based on a talk presented by the first author at the Short Program on Riemannian Geometry that took place at the Centre de Recherche Mathématiques, Université de Montréal, during the period June 28-July 16, 2004. It is a report on our joint work with János Kollár concerning the existence of an abundance of Einstein metrics on odd dimensional spheres, including exotic spheres.
The purpose of this note is to introduce a new method for proving the existence of Sasakian-Einstein metrics on certain simply connected odd dimensional manifolds. We then apply this method to prove the existence of new Sasakian-Einstein metrics on $\scriptstyle{S^2\times S^3}$ and on $\scriptstyle{(S^2\times S^3)# (S^2\times S^3).}$ These give the first known examples of non-regular Sasakian-Einstein 5-manifolds. Our method involves describing the Sasakian-Einstein structures as links of certain isolated hypersurface singularities, and makes use of the recent work of Demailly and Kollár, AG/9…
Extending the work of G. Székelyhidi and T. Brönnle to Sasakian manifolds we prove that a small deformation of the complex structure of the cone of a constant scalar curvature Sasakian manifold admits a constant scalar curvature structure if it is K-polystable. This also implies that a small deformation of the complex structure of the cone of a constant scalar curvature structure is K-semistable. As applications we give examples of constant scalar curvature Sasakian manifolds which are deformations of toric examples, and we also show that if a 3-Sasakian manifold admits a non-trivial transvers…
In this paper, we show that the uniform L^{4}-bound of the transverse Ricci curvature along the Sasaki-Ricci flow on a compact quasi-regular Sasakian (2n+1)-manifold M of general type. As an application, any solution of the normalized Sasaki-Ricci flow converges in the Cheeger-Gromov sense to the unique singular Sasaki η-Einstein metric on the transverse canonical model M_{can} of M if n is less than or equal to 3. In particular for n equal to 2, M_{can} is a S^{1}-orbibundle over the unique Keahler-Einstein orbifold surface (Z_{can},ω_{KE}) with finite point orbifold singularities. The floati…
In this note, we construct new examples of Lorentzian Sasaki-Einstein (LSE) metrics on Smale manifolds $M.$ It has already been established in \cite{Gmz2} that such metrics exist on the so-called torsion free Smale manifolds, i.e. the $k$-fold connected sum of $S^{2}\times S^{3}.$ Now, we show that LSE metrics exist on Smale manifolds for which $H_{2}(M,\mathbb{Z})_{tor}$ is nontrivial. In particular, we show that most simply-connected positive Sasakian rational homology 5-spheres are also negative Sasakian (hence Lorentzian Sasaki-Einstein). Moreover, we show that for each pair of positive in…
Let $G$ be a complex, connect reductive Lie group which is the complexification of a compact Lie group $K$. Let $M$ be a $\mathbb Q$-Fano $G$-compactification. In this paper, we first prove the uniqueness of $K\times K$-invariant (singular) Kähler-Einstein metric. Then we show the existence of (singular) Kähler-Einstein metric implies properness of the reduced Ding functional. Finally, we show that the barycenter condition is also necessary of properness.
On a sub-Riemannian manifold, a connection with skew-symmetric torsion is defined as the unique connection from the class of $N$-connections that has this property. Two cases are considered separately: sub-Riemannian structure of even rank, and sub-Riemannian structure of odd rank. The resulting connection, called the canonical connection, is not a metric connection in the case when the sub-Riemannian structure is of even rank. The structure of an almost quasi-Sasakian manifold is defined as an almost contact metric structure of odd rank that satisfies additional requirements. Namely, it is re…
We give a complete criterion for the existence of generalized Kähler Einstein metrics on toric Fano manifolds from view points of a uniform stability in a sense of GIT and the properness of a functional on the space of Kähler metrics.
We verify the extension to the zero section of momentum construction of Kaehler-Einstein metrics and Kaehler-Ricci solitons on the total space Y of positive rational powers of the canonical line bundle of toric Fano manifolds with possibly irregular Sasaki-Einstein metrics. More precisely, we show that the extended metric along the zero section has an expression which can be extended to Y, restricts to the associated unit circle bundle as a transversely Kaehler-Einstein (Sasakian eta-Einstein) metric scaled in the Reeb flow direction, and that there is a Riemannian submersion from the scaled S…
Let $(X,ω)$ be a compact normal Kähler space, with Hodge metric $ω$. In this paper, the last in a sequence of works studying the relationship between energy properness and canonical Kähler metrics, we introduce a geodesic metric structure on $\mathcal H_ω(X)$, the space of Kähler potentials, whose completion is the finite energy space $\mathcal E^1_ω(X)$. Using this metric structure and the results of Berman-Boucksom-Eyssidieux-Guedj-Zeriahi as ingredients in the existence/properness principle of Rubinstein and the author, we show that existence of Kähler-Einstein metrics on log Fano pairs is …
Many authors have studied Ricci solitons and their analogs within the framework of (almost) contact geometry. In this article, we thoroughly study the $(m,ρ)$-quasi-Einstein structure on a contact metric manifold. First, we prove that if a $K$-contact or Sasakian manifold $M^{2n+1}$ admits a closed $(m,ρ)$-quasi-Einstein structure, then it is an Einstein manifold of constant scalar curvature $2n(2n+1)$, and for the particular case -- a non-Sasakian $(k,μ)$-contact structure -- it is locally isometric to the product of a Euclidean space $\RR^{n+1}$ and a sphere $S^n$ of constant curvature $4$. …
This is an expository paper describing the geometry of certain Sasakian-Einstein manifolds. Such manifolds have recently become of interest due to Maldacena's AdS/CFT conjecture. They describe near-horizon geometries of branes at conical singularities.
In this paper, we prove an existence theorem of a local moduli space for geometric structures in a very general setting. Then to show the interest of this result, we apply it to the case of sasakian and Sasaki-Einstein structures.
In this expository article we discuss the relations between Sasakian geometry, reduced holonomy and supersymmetry. It is well known that the Riemannian manifolds other than the round spheres that admit real Killing spinors are precisely Sasaki-Einstein manifolds, 7-manifolds with a nearly parallel G2 structure, and nearly Kaehler 6-manifolds. We then discuss the relations between the latter two and Sasaki-Einstein geometry.
