Sasaki Metric Research Articles

Sasakian geometry on sphere bundles II: Constant scalar curvature

In a previous paper [18] the authors employed the fiber join construction of Yamazaki [38] together with the admissible construction of Apostolov, Calderbank, Gauduchon, and Tønnesen-Friedman [2] to construct new extremal Sasaki metrics on odd dimensional sphere bundles over smooth projective algebraic varieties. In the present paper we continue this study by applying a recent existence theorem [14] that shows that under certain conditions one can always obtain a constant scalar curvature Sasaki metric in the Sasaki cone. Moreover, we explicitly describe this construction for certain sphere bundles of dimension 5 and 7.

Twisted Sasaki metric on the unit tangent bundle and harmonicity

Twisted Sasaki metric on the unit tangent bundle and harmonicity

L-INTEGRAL FOR ISOPERIMETRIC EXTREMALS OF ROTATION

In this paper, the geodetic flux on a spherical tangent bundle of a two-dimensional Riemannian manifold with the Sasaki metric is considered and it is shown that, if the basis manifold is locally isometric of the surface of rotation, then the Hamiltonian system corresponding to the flow is fully integrated according to Liouville. Hence, as a consequence, the flow trajectories are in quadratures.

Weighted K–stability and coercivity with applications to extremal Kähler and Sasaki metrics

Weighted K–stability and coercivity with applications to extremal Kähler and Sasaki metrics

On the Geometry of Kobayashi–Nomizu Type and Yano Type Connections on the Tangent Bundle with Sasaki Metric

In this paper, we address the study of the Kobayashi–Nomizu type and the Yano type connections on the tangent bundle TM equipped with the Sasaki metric. Then, we determine the curvature tensors of these connections. Moreover, we find conditions under which these connections are torsion-free, Codazzi, and statistical structures, respectively, with respect to the Sasaki metric. Finally, we introduce the mutual curvature tensor on a manifold. We investigate some of its properties; furthermore, we study mutual curvature tensors on a manifold equipped with the Kobayashi–Nomizu type and the Yano type connections.

Existence and non-existence of constant scalar curvature and extremal Sasaki metrics

Existence and non-existence of constant scalar curvature and extremal Sasaki metrics

Some Results on Tangent Bundles with Berger Type Deformed Sasaki Metric over Kählerian Manifolds

Some Results on Tangent Bundles with Berger Type Deformed Sasaki Metric over Kählerian Manifolds

On the Geometry of Tangent Bundle and Unit Tangent Bundle with Deformed-Sasaki Metric

Let $(M^{m}, g)$ be a Riemannian manifold and $TM$ its tangent bundle equipped with a deformed Sasaki metric. In this paper, firstly we investigate all forms of Riemannian curvature tensors of $TM$ (Riemannian curvature tensor, Ricci curvature, sectional curvature and scalar curvature). Secondly, we study the geometry of unit tangent bundle equipped with a deformed Sasaki metric, where we presented the formulas of the Levi-Civita connection and also all formulas of the Riemannian curvature tensors of this metric.

Pseudo-Kähler and pseudo-Sasaki structures on Einstein solvmanifolds

The aim of this paper is to construct left-invariant Einstein pseudo-Riemannian Sasaki metrics on solvable Lie groups. We consider the class of mathfrak {z}-standard Sasaki solvable Lie algebras of dimension 2n+3, which are in one-to-one correspondence with pseudo-Kähler nilpotent Lie algebras of dimension 2n endowed with a compatible derivation, in a suitable sense. We characterize the pseudo-Kähler structures and derivations giving rise to Sasaki–Einstein metrics. We classify mathfrak {z}-standard Sasaki solvable Lie algebras of dimension le 7 and those whose pseudo-Kähler reduction is an abelian Lie algebra. The Einstein metrics we obtain are standard, but not of pseudo-Iwasawa type.

The $S^3_{\boldsymbol{w}}$ Sasaki join construction

The main purpose of this work is to generalize the $S^{3}_{\boldsymbol{w}}$ Sasaki join construction $M \star_{\boldsymbol{l}} S^{3}_{\boldsymbol{w}}$ described in the authors' 2016 paper when the Sasakian structure on $M$ is regular, to the general case where the Sasakian structure is only quasi-regular. This gives one of the main results, Theorem 3.2, which describes an inductive procedure for constructing Sasakian metrics of constant scalar curvature. In the Gorenstein case ($c_{1}(\mathscr{D}) = 0$) we construct a polynomial whose coeffients are linear in the components of $\boldsymbol{w}$ and whose unique root in the interval $(1, \infty)$ completely determines the Sasaki–Einstein metric. In the more general case we apply our results to prove that there exists infinitely many smooth 7-manifolds each of which admit infinitely many inequivalent contact structures of Sasaki type admitting constant scalar curvature Sasaki metrics (see Corollary 6.15). We also discuss the relationship with a recent paper of Apostolov and Calderbank as well as the relation with K-stability.

Notes on the second-order tangent bundles with the deformed Sasaki metric

The paper deals with the second-order tangent bundle T2MT2M with the deformed Sasaki metric ¯gg¯ over an n−dimensional Riemannian manifold gover an n−dimensional Riemannian manifold (M,g)(M,g). We calculate all Riemannian curvature tensor fields of the deformed Sasaki metric ¯gg¯ and search Einstein property of and search Einstein property of T2MT2M. Also the weakly symmetry properties of the deformed Sasaki metric are presented.

Sasakian metrics as generalized $$\eta $$-Ricci soliton

Sasakian metrics as generalized $$\eta $$-Ricci soliton

Inhomogeneous generalization of Einstein’s static universe with Sasakian space

Abstract We construct exact static inhomogeneous solutions to Einstein’s equations with counter flow of particle fluid and a positive cosmological constant by using the Sasaki metrics on three-dimensional spaces. The solutions, which admit an arbitrary function that denotes the inhomogeneous number density of particles, are a generalization of Einstein’s static universe. For some examples of explicit solutions, we discuss non-linear density contrast and deviation of the metric functions.

Ricci–Yamabe solitons on a class of generalized Sasakian space forms

In this paper, we characterize Ricci–Yamabe solitons and gradient Ricci–Yamabe solitons on 3-dimensional generalized Sasakian space forms with quasi Sasakian metric. Furthermore, we study [Formula: see text]-Ricci–Yamabe solitons and gradient [Formula: see text]-Ricci–Yamabe solitons on 3-dimensional generalized Sasakian space forms with quasi Sasakian metric. Finally, we construct an example to verify a result of our paper.

Emergent behaviors of the discrete thermodynamic Cucker–Smale model on complete Riemannian manifolds

We propose an intrinsic discrete-time counterpart of the abstract thermomechanical Cucker–Smale (TCS) model on connected, complete, and smooth Riemannian manifolds and study its emergent dynamics. Our proposed discrete model is expressed in terms of exponential map on the tangent bundle endowed with the Sasaki metric. Compared to projection-based discrete models on the manifold, it is embedding free and enjoys the same structural properties as the corresponding continuous models. For the proposed model, we provide a sufficient framework leading to asymptotic velocity alignment in which all particles’ velocity align when they lie in the same tangent plane via the parallel transport along the length-minimizing geodesic. For the unit-d sphere (Sd), we provide explicit representations of the Sasaki metric and the corresponding geodesics on TSd and show that the TCS model exhibits a dichotomy in asymptotic spatial patterns (either energy tends to zero or all particles move along a common geodesic on Sd, which is a great circle). We also provide several numerical examples and compare them with analytical results.

Some Notes on Berger Type Deformed Sasaki Metric in the Cotangent Bundle

In the present paper, we study some notes on Berger type deformed Sasaki metric in the cotangent bundle T∗MT∗M over an anti-paraKähler manifold (M,φ,g)(M,φ,g). We characterize some geodesic properties for this metric. Next we also construct some almost anti-paraHermitian structures on T∗MT∗M and search conditions for these structures to be anti-paraKähler and quasi-anti-paraKähler with respect to the Berger type deformed Sasaki metric.

C1,1 regularity of degenerate complex Monge–Ampère equations and some applications

In this paper, we prove a $C^{1,1}$ estimate for solutions of complex Monge-Amp\`{e}re equations on compact almost Hermitian manifolds. Using this $C^{1,1}$ estimate, we show existence of $C^{1,1}$ solutions to the degenerate Monge-Amp\`{e}re equations, the corresponding Dirichlet problems and the singular Monge-Amp\`{e}re equations. We also study the singularities of the pluricomplex Green's function. In addition, the proof of the above $C^{1,1}$ estimate is valid for a kind of complex Monge-Amp\`{e}re type equations. As a geometric application, we prove the $C^{1,1}$ regularity of geodesics in the space of Sasakian metrics.

Weighted K-stability of polarized varieties and extremality of Sasaki manifolds

We use the correspondence between extremal Sasaki structures and weighted extremal Kähler metrics defined on a regular quotient of a Sasaki manifold, established by the first two authors in [1], and the theory of weighted K-stability (introduced by Lahdili in [39]) in order to define a suitable notion of (relative) weighted K-stability for compact Sasaki manifolds of regular type. We show that the (relative) weighted K-stability with respect to a maximal torus and smooth equivariant test configurations with reduced central fibre is a necessary condition for the existence of a (possibly irregular) extremal Sasaki metric. We also compare weighted K-stability to the K-stability of the corresponding polarized affine cone (introduced by Collins–Székelyhidi in [18]), and prove that they agree on the class of test configurations we consider. As a byproduct, we strengthen the obstruction to the existence of a scalar-flat Kähler cone metric found in [18] from the K-semistability to the K-stability on these test configurations. We use our approach to give a characterization of the existence of a compatible extremal Sasaki structure on a principal S1-bundle over an admissible ruled manifold in the sense of [4], expressed in terms of the positivity of a single polynomial of one variable over a given interval.

Para-complex structures on linear coframe bundle with Sasakian metric

Para-complex structures on linear coframe bundle with Sasakian metric

Some Biharmonic Problems on the Tangent Bundle with a Berger-type Deformed Sasaki Metric

Let (M<sub>2k</sub>,Φ,g) be an almost anti-paraKahler manifold and TM its tangent bundle equipped with the Berger type deformed Sasaki metric gBS and the paracomplex structure Φ<sub>˜</sub>. In this paper, we deal with the biharmonicity of canonical projection π : TM →M and a vector field X which is considered as a map X : M → TM.