Researchers found all invariant contact structures on tangent sphere bundles of compact symmetric spaces.
problem Identifying all invariant contact metric structures on tangent sphere bundles of compact rank-one symmetric spaces.
method Explicitly obtained all structures, distinguishing K-contact, Sasakian, and 3-Sasakian structures.
result There is a unique Sasakian-Einstein metric on tangent sphere bundles of spheres and real projective spaces.
New proof confirms nearly Sasakian condition for high-dimensional manifolds.
problem Characterizing Sasakian manifolds among almost contact metric manifolds.
method Self-contained, conceptual proof for nearly Sasakian condition.
result An almost contact metric manifold is Sasakian if and only if it is nearly Sasakian.
Study on submanifolds in generalized Sasakian-space-forms with various connections.
problem Analyzing submanifolds in generalized Sasakian-space-forms with different connections.
method Examines submanifolds in generalized Sasakian-space-forms with semisymmetric metric, non-metric, Schouten-van Kampen, and Tanaka-webster connections.
result Provides results on submanifolds in generalized Sasakian-space-forms with respect to various connections.
Deformations of the Reeb flow of a Sasakian manifold as transversely Kähler flows may not admit compatible Sasakian metrics anymore. We show that the triviality of the (0,2)-component of the basic Euler class characterizes the existence of compatible Sasakian metrics for given small deformations of the Reeb flow as tra…
Complex structures found on product of Sasakian manifolds.
problem Exploring complex structures on the product of two Sasakian manifolds.
method Examined a family of complex structures on the product of two compact Sasakian manifolds, computed Dolbeault cohomology groups.
result None of the complex structures in the family admit compatible locally conformally Kähler metrics if both Sasakian manifolds are of dimension greater than 1.
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 man…
Geometric flows study nearly parallel G2-structures on 3-Sasakian 7-manifolds.
problem Analyzing geometric flows of G2-structures on 3-Sasakian manifolds.
method Study of Laplacian flow and Laplacian coflow of G2-structures on 3-Sasakian manifolds.
result Distinct behavior of flows, notably regarding stability of nearly parallel G2-structures.
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 i…
Using the Hard Lefschetz Theorem for Sasakian manifolds, we find two examples of compact K-contact nilmanifolds with no compatible Sasakian metric in dimensions five and seven, respectively
In this paper, we prove that the transverse Mabuchi K-energy functional is convex along the weak geodesic in the space of Sasakian metrics. As an application, we obtain the uniqueness of constant scalar curvature Sasakian metrics modulo automorphisms for the transverse holomorphic structure.
Defines a new Poisson structure for generalized Sasakian spaces.
problem No specific problem stated; focuses on new structure definition.
method Defines a canonical Poisson structure on generalized contact metric spaces.
result Shows distinction between generalized Sasakian and coKähler structures.
We study the space of Sasaki metrics on a compact manifold M by introducing an odd-dimensional analogue of the J-flow. That leads to the notion of critical metric in the Sasakian context. In analogy to the Kähler case, on a polarised Sasakian manifold there exists at most one normalised critical metric. The flow is…
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 …
Study invariant and anti-invariant submanifolds in special quasi-Sasakian manifolds.
problem Characterize invariant and anti-invariant submanifolds in special quasi-Sasakian manifolds.
method Investigate Chaki-pseudo parallel and Deszcz-pseudo parallel invariant submanifolds with Levi-Civita and semisymmetric metric connections. Also study invariant and anti-invariant submanifolds with Ricci solitons.
result Invariant and anti-invariant submanifolds are equivalent under certain conditions.
Study shows Sasakian manifolds with non-negative Ricci curvature have synthetic lower bounds.
problem Synthetic Ricci curvature bounds on Sasakian manifolds.
method Used measure contraction property and synthetic Ricci curvature lower bounds.
result Sasakian manifolds with non-negative Ricci curvature satisfy measure contraction property.
Study of a basic Hitchin equation on Sasakian 3-folds, showing hyperKähler metric.
problem Understanding moduli spaces of the basic Hitchin equation on Sasakian 3-folds.
method Construction of moduli space and calculation of its dimension.
result Moduli space admits a hyperKähler metric.
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.
The paper characterizes Sasakian manifolds using weak nearly Sasakian structures.
problem Characterizing Sasakian manifolds using a new structure.
method Study of weak nearly Sasakian structures and proving theorems.
result Provides a new criterion for a weak almost contact metric manifold to be Sasakian.
The study of *-Ricci solitons on para-Sasakian manifolds.
problem Characterizing para-Sasakian manifolds with *-Ricci solitons.
method Analyzing the conditions under which a para-Sasakian metric becomes a *-Ricci soliton.
result Para-Sasakian manifolds that are *-Ricci solitons are either D-homothetic to Einstein manifolds or have vanishing Ricci tensor.
Study on para-Sasakian metrics and their solitons.
problem Characterizing para-Sasakian metrics with conformal η-Ricci solitons.
method Analyzing the properties of para-Sasakian metrics under conformal η-Ricci solitons.
result Para-Sasakian metrics admitting conformal η-Ricci solitons are η-Einstein.
New insights into 5D Sasakian Lie algebras with trivial center.
problem Characterizing five-dimensional Sasakian Lie algebras with trivial center.
method Analyzing φ-symmetry and constructing contact metric structures. result Every 5D Sasakian Lie algebra with trivial center is φ-symmetric. New metric structures generalize Sasakian and cosymplectic structures, proving rigidity and finding conditions.
problem Generalizing Sasakian and cosymplectic structures to new metric structures.
method Introducing weak structures and proving rigidity of Sasakian structures.
result Any weak Sasakian structure is homothetically equivalent to a Sasakian structure.
New criteria for Sasakian manifolds and classification of nearly cosymplectic manifolds.
problem Characterizing Sasakian and cosymplectic manifolds.
method Analyzing nearly Sasakian and nearly cosymplectic manifolds of dimensions greater than five.
result Every nearly Sasakian manifold of dimension greater than five is Sasakian.
The study introduces a new soliton concept to classify Sasakian 3-manifolds.
problem Classifying Sasakian 3-manifolds under specific conditions.
method Introducing and studying ∗-Ricci-Yamabe solitons on contact metric manifolds. result Sasakian 3-manifolds admitting ∗-Ricci-Yamabe solitons are ∗-Ricci flat, positive Sasakian, and have Fano transverse geometry. The question of whether a Sasakian metric can admit an additional compatible (K-)contact structure is addressed. In the complete case if the second structure is also assumed Sasakian, works of Tachibana-Yu and Tanno show that the manifold must be 3-Sasakian or an odd dimensional sphere with constant curvature. Some ext…
The paper examines *-Ricci tensor properties on Sasakian manifolds.
problem Analyzing *-Ricci tensor on Sasakian manifolds.
method Examining φ-confomally flat and confomally flat *-η-Einstein Sasakian manifolds, *-Ricci symmetric condition, and *-Ricci soliton.
result Investigates properties of *-Ricci tensor on Sasakian manifolds.
This paper is devoted to the regularity analysis of a geodesic equation in the space of Sasakian metrics. Firstly, we reduce the geodesic equation in the space of Sasakian metrics to a Dirichlet problem of degenerate complex Monge-Ampére type eqution on the Kähler cone; secondly, we obtain a priori etimates for the abo…
Study on new connections in para-Sasakian manifolds.
problem Exploring new generalized symmetric metric connections in para-Sasakian manifolds.
method Identified and utilized generalized symmetric connections of type (α,β) on Lorentzian para-Sasakian manifolds.
result Obtained new results involving curvature and Ricci tensors on para-Sasakian manifolds.
The paper defines a new connection on sub-Riemannian manifolds and explores conditions for almost quasi-Sasakian manifolds to be Einstein.
problem Exploring new connections and conditions for specific types of manifolds.
method Defining a skew-symmetric connection and studying its properties on sub-Riemannian manifolds, and examining conditions for almost quasi-Sasakian manifolds to be Einstein.
result Sufficient conditions are found for an almost quasi-Sasakian manifold to be an Einstein manifold.
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 …
Study weak quasi contact metric manifolds to generalize K-contact and Sasakian manifolds criteria.
problem Generalize K-contact and Sasakian manifolds criteria using weak quasi contact metric manifolds.
method Study weak quasi contact metric manifolds and generalize theorems for K-contact and Sasakian manifolds.
result Provide new criterions for K-contact and Sasakian manifolds in terms of curvature tensor and geometric objects.
Kähler cones over Sasakian manifolds are flat if projectively induced.
problem Characterizing Kähler cones over Sasakian manifolds.
method Relating Kähler potentials and using Ricci-flatness.
result Kähler cones over regular Sasakian manifolds are flat if projectively induced.
Survey of Sasakian geometry and its moduli.
problem Understanding Sasakian moduli.
method Survey and analysis of Sasakian structures.
result Exploration of Sasaki-Einstein metrics moduli.
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…
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…
Study on a specific type of Lie algebras with Kähler and contact properties.
problem Characterizing and classifying transversely Kähler almost contact metric Lie algebras.
method Analyzing properties of Lie algebras with contact forms and Kähler structures, considering center dimensions and quotient properties.
result Classification of 5-dimensional η-Einstein transversely Kähler almost contact metric Lie algebras.
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 $-…
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.
Study on Legendre curves in non-Sasakian manifolds with curvature properties.
problem Characterizing Legendre curves in non-Sasakian contact metric manifolds.
method Analyzing C-parallel and C-proper mean curvature vector fields. result Curvature characterizations of Legendre curves in non-Sasakian manifolds.
The study proves estimates for transverse nonlinear equations on Sasakian manifolds with applications in geometry.
problem Estimating transverse fully nonlinear equations on Sasakian manifolds.
method Proving a priori estimates for transverse fully nonlinear equations.
result The study proves estimates for transverse fully nonlinear equations on Sasakian manifolds and gives geometric applications.
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.
Study proves infinite isometry groups for certain Sasakian manifolds.
problem Understanding the isometry groups of Sasakian manifolds.
method Proves existence of non-isometric metrics leading to infinite fundamental groups.
result Fundamental groups of isometry groups are infinite for positive ρ.
Study invariant submanifolds in generalized Sasakian-space-forms.
problem Characterize invariant submanifolds in generalized Sasakian-space-forms.
method Analyzes invariant submanifolds with respect to Levi-Civita and semi-symmetric metric connections.
result Provides necessary and sufficient conditions for total geodesic submanifolds.
We study the Lie algebra of infinitesimal isometries on compact Sasakian and K--contact manifolds. On a Sasakian manifold which is not a space form or 3--Sasakian, every Killing vector field is an infinitesimal automorphism of the Sasakian structure. For a manifold with K--contact structure, we prove that there exists …
Study properties of 3D trans-Sasakian manifolds with η-Yamabe solitons.
problem Properties of 3D trans-Sasakian manifolds with η-Yamabe solitons.
method Investigated curvature conditions and constructed a manifold.
result Construct a 3D trans-Sasakian manifold satisfying η-Yamabe soliton.
Study of 3D trans-Sasakian manifolds using Newman--Penrose formalism.
problem Characterizing and understanding the geometry of 3D trans-Sasakian manifolds.
method Using Newman--Penrose formalism to encode the geometry of the structure vector field.
result Derivation of curvature and Laplacian identities for trans-Sasakian manifolds and their subclasses, including rigidity 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 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-Einste…
New class of manifolds studied with special properties.
problem Characterizing and studying a new class of almost contact metric manifolds.
method Introducing anti-quasi-Sasakian manifolds and analyzing their properties.
result Anti-quasi-Sasakian manifolds with constant curvature are flat and cokähler.