Sasaki Metric Research Articles

Sasakian metric as a Ricci soliton and related results

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 us to identify the Sasakian metric on the Heisenberg group H2n+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.

Variations of gwistor space

We study natural variations of the \mathrm{G}_2 structure \sigma_0\in\Lambda^3_+ existing on the unit tangent sphere bundle SM of any oriented Riemannian 4-manifold M . We find a circle of structures for which the induced metric is the usual one, the so-called Sasaki metric, and prove how the original structure has a preferred role in the theory. We deduce the equations of calibration and cocalibration, as well as those of W_3 pure type and nearly-parallel type.

The Sasaki–Ricci Flow on Sasakian 3-Spheres

We show that on a Sasakian 3-sphere the Sasaki–Ricci flow initiating from a Sasakian metric of positive transverse scalar curvature converges to a gradient Sasaki–Ricci soliton. We also show the existence and uniqueness of gradient Sasaki–Ricci soliton on each Sasakian 3-sphere.

Extremal Sasakian horizons

We point out a simple construction of an infinite class of Einstein near-horizon geometries in all odd dimensions greater than five. Cross-sections of the horizons are inhomogeneous Sasakian metrics (but not Einstein) on S3×S2 and more generally on S3-bundles over any compact positive Kähler–Einstein manifold. They are all consistent with the known topology and symmetry constraints for asymptotically flat or globally AdS black holes.

Sasaki Metrics for Analysis of Longitudinal Data on Manifolds.

Longitudinal data arises in many applications in which the goal is to understand changes in individual entities over time. In this paper, we present a method for analyzing longitudinal data that take values in a Riemannian manifold. A driving application is to characterize anatomical shape changes and to distinguish between trends in anatomy that are healthy versus those that are due to disease. We present a generative hierarchical model in which each individual is modeled by a geodesic trend, which in turn is considered as a perturbation of the mean geodesic trend for the population. Each geodesic in the model can be uniquely parameterized by a starting point and velocity, i.e., a point in the tangent bundle. Comparison between these parameters is achieved through the Sasaki metric, which provides a natural distance metric on the tangent bundle. We develop a statistical hypothesis test for differences between two groups of longitudinal data by generalizing the Hotelling T 2 statistic to manifolds. We demonstrate the ability of these methods to distinguish differences in shape changes in a comparison of longitudinal corpus callosum data in subjects with dementia versus healthily aging controls.

A 3-DIMENSIONAL SASAKIAN METRIC AS A YAMABE SOLITON

If a 3-dimensional Sasakian metric on a complete manifold (M, g) is a Yamabe soliton, then we show that g has constant scalar curvature, and the flow vector field V is Killing. We further show that, either M has constant curvature 1, or V is an infinitesimal automorphism of the contact metric structure on M.

TOTALLY GEODESIC SUBMANIFOLDS OF GL-MANIFOLDS

In this paper, for a Finsler manifold (M, F) with a Finsler metric gij(x, y) we shall consider a generalized Lagrange metrics (FGL-metrics) as the form *gij(x, y) = gij(x, y) + σ(x, y)Bi(x, y)Bj(x, y) on TM. Then we shall consider a Riemannian manifold (TM, *G) in which *G is a generalized Sasakian metric of *g on [Formula: see text]. Then we restrict the above FGL-metrics to a submanifold of [Formula: see text], and show that it admits a GL-metric structure. Then we shall find a necessary and sufficient condition for this submanifold to be totally geodesic.

Geodesicity and isoclinity properties for the tangent bundle of the Heisenberg manifold with Sasaki metric

We prove that the horizontal and vertical distributions of the tangent bundle with the Sasaki metric are isocline, the distributions given by the kernels of the horizontal and vertical lifts of the contact form \omega on the Heisenberg manifold (H_3,g) to (TH_3,g^S) are not totally geodesic, and the distributions F^H=L(E_1^H,E_2^H) and F^V=L(E_1^V,E_2^V) are totally geodesic, but they are not isocline. We obtain that the horizontal and natural lifts of the curves from the Heisenberg manifold (H_3,g), are geodesics on the tangent bundle endowed with the Sasaki metric (TH_3,g^s), if and only if the curves considered on the base manifold are geodesics. Then, we get two particular examples of geodesics on (TH_3,g^s), which are not horizontal or natural lifts of geodesics from the base manifold (H_3,g).

Rayleigh dissipation from the general recurrence of metrics in path spaces

Given a pair (metric g, symmetric 2-covariant tensor field H though as a Rayleigh dissipation) on a path space (manifold M, semispray S), the family of nonlinear connections N such that H equals the dynamical derivative of g with respect to (S,N) is determined by using the Obata tensors. In this way, we generalize the case of metric nonlinear connections as well as that of recurrent metrics. As applications, we treat firstly the case of Finslerian (α,β)-metrics finding all nonlinear connections for which the associated Finsler–Sasaki metric is exactly the dynamical derivative of the Riemannian–Sasaki metric. Secondly, we apply our results for the case of Beil metrics used in Relativity and field theories.

Regularity of the geodesic equation in the space of Sasakian metrics

In this paper, we study a geodesic equation in the space of Sasakian metrics H. The equation leads to the Dirichlet problem of a complex Monge–Ampère type equation on the Kähler cone. This equation differs from the standard complex Monge–Ampère equation in a significant way, with gradient terms involved in the (1,1) symmetric tensor of the operator. We establish appropriate regularity estimates for this complex Monge–Ampère type equation. As geometric application, we show that the space of Sasakian metrics H is a metric space, and the constant transversal scalar curvature metric realizes the global minimum of K-energy if the first basic Chern class C1B⩽0. We also prove that the constant transversal scalar curvature metric is unique in each basic Kähler class if the first basic Chern class is either strictly negative or zero.

Energy functionals and canonical metrics in Sasakian geometry

In this paper, we introduce E k functionals in Sasakian geometry, and deduce their Euler-Lagrange equations. Then, we prove the uniqueness results for critical metrics of these functionals. We also obtain some uniqueness results for Sasakian metrics with constantly transverse σ k curvature.

LargeNlimit of quiver matrix models and Sasaki-Einstein manifolds

We study the matrix models that result from the localization of the partition functions of $\mathcal{N}=2$ Chern-Simons--matter theories on the three-sphere. A large class of such theories are conjectured to be holographically dual to M-theory on Sasaki-Einstein seven-manifolds. We study the M-theory limit (large $N$ and fixed Chern-Simons levels) of these matrix models for various examples, and show that in this limit the free energy reproduces the expected AdS/CFT result of ${N}^{3/2}/\mathrm{Vol}(Y{)}^{1/2}$, where $\mathrm{Vol}(Y)$ is the volume of the corresponding Sasaki-Einstein metric. More generally we conjecture a relation between the large $N$ limit of the partition function, interpreted as a function of trial $R$ charges, and the volumes of Sasakian metrics on links of Calabi-Yau fourfold singularities. We verify this conjecture for a family of $U(N{)}^{2}$ Chern-Simons quivers based on M2 branes at hypersurface singularities, and for a $U(N{)}^{3}$ theory based on M2 branes at a toric singularity.

Existence and non-uniqueness of constant scalar curvature toric Sasaki metrics

AbstractWe study compatible toric Sasaki metrics with constant scalar curvature on co-oriented compact toric contact manifolds of Reeb type of dimension at least five. These metrics come in rays of transversal homothety due to the possible rescaling of the Reeb vector fields. We prove that there exist Reeb vector fields for which the transversal Futaki invariant (restricted to the Lie algebra of the torus) vanishes. Using an existence result of E. Legendre [Toric geometry of convex quadrilaterals, J. Symplectic Geom. 9 (2011), 343–385], we show that a co-oriented compact toric contact 5-manifold whose moment cone has four facets admits a finite number of rays of transversal homothetic compatible toric Sasaki metrics with constant scalar curvature. We point out a family of well-known toric contact structures on S2×S3 admitting two non-isometric and non-transversally homothetic compatible toric Sasaki metrics with constant scalar curvature.

Towards the F-theorem: $ \mathcal{N} = 2 $ field theories on the three-sphere

For 3-dimensional field theories with {\cal N}=2 supersymmetry the Euclidean path integrals on the three-sphere can be calculated using the method of localization; they reduce to certain matrix integrals that depend on the R-charges of the matter fields. We solve a number of such large N matrix models and calculate the free energy F as a function of the trial R-charges consistent with the marginality of the superpotential. In all our {\cal N}=2 superconformal examples, the local maximization of F yields answers that scale as N^{3/2} and agree with the dual M-theory backgrounds AdS_4 x Y, where Y are 7-dimensional Sasaki-Einstein spaces. We also find in toric examples that local F-maximization is equivalent to the minimization of the volume of Y over the space of Sasakian metrics, a procedure also referred to as Z-minimization. Moreover, we find that the functions F and Z are related for any trial R-charges. In the models we study F is positive and decreases along RG flows. We therefore propose the "F-theorem" that we hope applies to all 3-d field theories: the finite part of the free energy on the three-sphere decreases along RG trajectories and is stationary at RG fixed points. We also show that in an infinite class of Chern-Simons-matter gauge theories where the Chern-Simons levels do not sum to zero, the free energy grows as N^{5/3} at large N. This non-trivial scaling matches that of the free energy of the gravity duals in type IIA string theory with Romans mass.

Curvatures of tangent hyperquadric bundles

We study geometry of tangent hyperquadric bundles over pseudo-Riemannian manifolds, which are equipped, as submanifolds of the tangent bundles, with the induced Sasaki metric. All kinds of curvatures are calculated, and geometric results concerning the Ricci curvature and the scalar curvature are proved. There exists a hyperquadric bundle whose scalar curvature is a preassigned constant.

On the geometry of the (1, 1)-tensor bundle with Sasaki type metric

Curvature properties are studied for the Sasaki metric on the (1, 1) tensor bundle of a Riemannian manifold. As an application, examples of almost para-Nordenian and para-Kahler-Nordenian B-metrics are constructed on the (1, 1) tensor bundle by looking at the Sasaki metric. Also, with respect to the para-Nordenian B-structure, paraholomorphic conditions for the complete lifts of vector fields are analyzed.

Curvatures of the tangent bundle with a special almost product metric

We study a class of Riemannian almost product metrics on the tangent bundle of a smooth manifold. This class includes the Sasaki and Cheeger-Gromoll metrics as special cases. For this class of metrics, we find the dependence of the scalar curvature of the tangent bundle on objects of the base manifold. For the case in which the base manifold is a space of constant sectional curvature, we obtain conditions on the metric and the dimension of the base under which the scalar curvature of the tangent bundle is constant. For special cases of metrics of the class considered, we find the intervals on which the scalar curvature of the tangent bundle treated as a function of the sectional curvature of the base has constant sign.

Tangent bundle of a hypersurface of a Euclidean space

We use the fact that the tangent bundle TM of an orientable hypersurface M in the Euclidean space R n+1 is a submanifold of the Euclidean space R 2n+2, and use the induced metric on TM as submanifold to study its geometry. This induced metric is not a natural metric in the sense that the projection π : TM → M is not a Riemannian submersion (which holds for Sasaki and other metrics, used to study geometry of the tangent bundle). First we prove that there is a reduction in the codimension of the submanifold TM and thus the tangent bundle TM is a hypersurface of the Euclidean space R 2n+1. As a consequence of our study, we infer that the induced metric on TS n the tangent bundle of the unit sphere S n makes TS n a Riemannian manifold of nonnegative sectional curvature. We also obtain a condition under which the tangent bundle TM of a hypersurface M in a Euclidean space is trivial.

SASAKIAN 3-MANIFOLD AS A RICCI SOLITON REPRESENTS THE HEISENBERG GROUP

We show that, if a 3-dimensional Sasakian metric is a non-trivial Ricci soliton, then it is expanding and homothetic to the standard Sasakian metric on the Heisenberg group nil3. We have also discussed properties of the Ricci soliton potential vector field that relate to the underlying contact structure.

Extremal Sasakian Metrics on ${\bf S^3}$-bundles over ${\bf S^2}$

Extremal Sasakian Metrics on ${\bf S^3}$-bundles over ${\bf S^2}$