The paper explores connections between projective Hessian and Sasakian manifolds.
problem Connecting projective Hessian and Sasakian geometries.
method Constructing Sasakian structures from projective Hessian structures on Lie groups.
result Construction of semi-Sasakian structures on Lie groups and groups of motions.
We obtain some general results on Sasakian Lie algebras and prove as a consequence that a (2n + 1)-dimensional nilpotent Lie group admitting left-invariant Sasakian structures is isomorphic to the real Heisenberg group H2n+1. Furthermore, we classify Sasakian Lie algebras of dimension 5 and determine which of th…
New method constructs degenerate Sasakian manifolds from hyperkähler bundles.
problem Constructing degenerate 3-(α,δ)-Sasakian manifolds. method Using fiber products of Boothby-Wang bundles over hyperkähler manifolds.
result No non-trivial compact examples exist, and one family of nilpotent Lie groups with this geometry is identified.
We show that every five-dimensional Sasakian Lie algebra with trivial center is φ-symmetric. Moreover starting from a particular Sasakian structure on the Lie group SL(2,R)×Aff(R) we obtain a family of contact metric (k,μ) structures whose Boeckx invariants assume all values le…
Study on invariant anti-quasi-Sasakian structures on compact manifolds.
problem Existence and classification of invariant anti-quasi-Sasakian structures of maximal rank.
method Analysis of invariant structures on compact homogeneous Riemannian manifolds and nilpotent Lie groups.
result Classification of invariant anti-quasi-Sasakian structures on nilpotent Lie groups.
Study invariant CKY 2-forms on 5D Lie groups, classifying and determining their properties.
problem Classifying and understanding CKY 2-forms on 5D Lie groups.
method Classification and analysis of 5D metric Lie algebras with CKY tensors.
result First examples of CKY 2-forms on metric Lie algebras without Sasakian structures.
Geometric quantization studied on CR manifolds with Lie group actions.
problem Quantization of CR manifolds with group actions.
method Study of CR Guillemin-Strernberg map and its Fredholm property for CR functions.
result Quantization commutes with reduction for Sasakian manifolds.
We study compact Sasakian manifolds whose Tondeur connection has holonomy group either trivial or contained in Sp(n). We show that the first condition forces the manifold to be a compact quotient of the Heisenberg Lie group, while in the simply-connected case a Sasakian structure has the holonomy of the Tondeur connect…
The study examines extensions of Lie algebras with specific geometric structures.
problem Conditions for preserving geometric structures in Lie algebra extensions.
method Analyzes extensions of Sasakian and Frobenius-Kähler Lie algebras.
result Conditions for maintaining Sasakian or Frobenius-Kähler structures after extensions.
We investigate some topological properties, in particular formality, of compact Sasakian manifolds. Answering some questions raised by Boyer and Galicki, we prove that all higher (than three) Massey products on any compact Sasakian manifold vanish. Hence, higher Massey products do obstruct Sasakian structures. Using th…
The paper reclassifies almost paracontact metric manifolds into 12 classes.
problem Classifying almost paracontact metric manifolds into distinct classes.
method Decomposing one class into two, determining various manifold types, and defining structures on Lie groups.
result 3-dimensional manifolds belong to four basic classes.
Study η-Einstein Sasakian structures on Lie algebras, dividing cases based on center dimension.
problem Investigate η-Einstein Sasakian structures on Lie algebras. method Divide cases based on center dimension, use theory of normal j-algebras and modifications of Hermitian Lie algebras to construct examples. result Construct new examples of η-Einstein Sasakian Lie algebras and solvmanifolds. 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.
We show that the Malcev Lie algebra of the fundamental group of a compact 2n+1-dimensional Sasakian manifold with n≥2 admits a quadratic presentation by using Morgan's bigradings of minimal models of mixed-Hodge diagrams. By using bigradings of minimal models, we also simplify the proof of the result of Cappelle…
Characterizes Vaisman solvmanifolds and connects them to other geometric structures.
problem Characterizing Vaisman solvmanifolds and their connections to other geometric structures.
method Characterization of unimodular solvable Lie algebras with Vaisman structures in terms of Kähler flat Lie algebras.
result Establishes algebraic restrictions for the existence of Vaisman structures and connects them to other geometric notions.
Classification results are given for (i) compact quaternionic Kähler manifolds with a cohomogeneity-one action of a semi-simple group, (ii) certain complete hyperKähler manifolds with a cohomogeneity-two action of a semi-simple group preserving each complex structure, (iii) compact 3-Sasakian manifolds which are cohomo…
In the present paper we carry on a systematic study of 3-quasi-Sasakian manifolds. In particular we prove that the three Reeb vector fields generate an involutive distribution determining a canonical totally geodesic and Riemannian foliation. Locally, the leaves of this foliation turn out to be Lie groups: either the o…
Proofs centerless unimodular contact Lie algebras.
problem Characterizing centerless unimodular contact Lie algebras.
method Elementary proof and introduction of DS-contact Lie algebras.
result The only centerless unimodular examples are sl(2,R) and su(2). The study characterizes Sasakian manifolds from magnetic Hopf surfaces.
problem Characterizing Sasakian manifolds from magnetic Hopf surfaces.
method Using a unit Killing vector field and Lie dragging a magnetic curve, the study characterizes Sasakian structures.
result If a magnetic Hopf surface is a constant mean curvature surface, then the manifold M is a Sasakian manifold.
New proof classifies homogeneous 3-Sasakian and quaternionic Kähler manifolds.
problem Classifying homogeneous 3-Sasakian and quaternionic Kähler manifolds.
method Constructing an explicit one-to-one correspondence via root systems and analyzing properties of real projective spaces.
result Derivation of the classification of homogeneous positive quaternionic Kähler manifolds.
Study Hodge theory and supersymmetry on Sasakian and Vaisman manifolds.
problem Decomposing harmonic forms and cohomology on Sasakian and Vaisman manifolds.
method Construct Lie superalgebra, use Hodge decomposition, compute supersymmetry algebra.
result Explicit computation of the supersymmetry algebra of Sasakian manifolds.
A contact 3-manifold M admitting a transversal Ricci soliton (g,v,λ) is either Sasakian or locally isometric to one of the Lie groups SU(2), SL(2,R), E(2), E(1,1) with a left invariant metric.
The study examines weakly Einstein metrics on specific types of manifolds.
problem Investigating weakly Einstein metrics on Sasakian and contact metric manifolds.
method Analyzing scalar curvature constraints and proving manifold properties.
result Classification and properties of weakly Einstein contact metric manifolds.
Study on invariant structures on 7D nilpotent Lie groups, focusing on Sasaki and K-contact.
problem Existence and classification of invariant Sasaki and K-contact structures on 7D nilpotent Lie groups.
method Analysis of 22 and 25 classes of 7D nilpotent Lie groups for pseudo-Sasaki and K-contact structures, respectively.
result Contact seven-dimensional algebras have nonzero Ricci tensor in contact directions.
Statistical Lie algebras with constant curvature are linked to locally conformally Kähler structures.
problem Characterizing Lie algebras with constant curvature and their geometric implications.
method Constructing statistical manifolds and Sasakian structures to relate Lie algebras to locally conformally Kähler structures.
result Statistical Lie algebras of constant curvature correspond to locally conformally Kähler Lie algebras.
Study of Sasakian immersions in Heisenberg group and BNimesR, focusing on η-Einstein case.
problem Classification of Sasakian immersions in specific spaces.
method Analysis of complete regular Sasakian manifolds in Heisenberg group and BNimesR with standard Sasakian structures. result Complete classification of η-Einstein Sasakian immersions. Proves one-relator groups are fundamental groups of Sasakian manifolds.
problem Characterizing groups that can be fundamental groups of Sasakian manifolds.
method Analyzes one-relator groups and classifies groups of deficiency at least two.
result Proves sufficient and necessary conditions for one-relator groups to be fundamental groups of Sasakian manifolds.
We show certain symmetry of the dimensions of cohomologies of the funda- mental groups of compact Sasakian manifolds by using the Hodge theory of twisted basic cohomology. As applications, we show that the polycyclic fundamental groups of compact Sasakian manifolds are virtually nilpotent and Sasakian solvmanifolds are…
We introduce the notion of even Clifford structures on Riemannian manifolds, a framework generalizing almost Hermitian and quaternion-Hermitian geometries. We give the complete classification of manifolds carrying parallel even Clifford structures: Kähler, quaternion-Kähler and Riemannian products of quaternion-Kähler …
Study of abelian structures on odd-dimensional Lie algebras and their geometric properties.
problem Characterizing abelian structures on odd-dimensional Lie algebras.
method Introducing and analyzing abelian almost contact and almost 3-contact structures, and their compatibility conditions.
result Classification of 5-dimensional Sasakian Lie algebras and 7-dimensional abelian almost 3-contact Lie algebras.
Study shows compact Sasakian manifolds are locally Heisenberg up to deformation.
problem Characterizing compact Sasakian manifolds.
method Analyzing basic Chern classes and using left invariant Sasakian structures.
result Compact Sasakian manifolds are locally isomorphic to the real Heisenberg group.
Nilpotent Sasakian manifolds are Heisenberg nilmanifolds.
problem Characterizing Sasakian manifolds with nilpotent fundamental groups.
method Proved diffeomorphism to Heisenberg nilmanifolds.
result Compact aspherical Sasakian manifolds with nilpotent fundamental groups are Heisenberg nilmanifolds.
We study the fundamental groups of compact Sasakian manifolds, which we call Sasaki groups. It is shown that all known Ka¨hler groups are Sasaki, in particular, all finite groups are Sasaki. On the other hand, we show there exists many restrictions on the fundamental groups of compact Sasakian manifolds. We als…
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.
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 …
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 extend the Donaldson-Corlette-Hitchin-Simpson correspondence between Higgs bundles and flat connections on compact Kähler manifolds to compact quasi-regular Sasakian manifolds. A particular consequence is the translation of restrictions on Kähler groups proved using the Donaldson-Corlette-Hitchin-Simpson corresponde…
Paper proves existence of nonconstant CR-holomorphic functions in Sasakian manifolds.
problem Proving existence of nonconstant CR-holomorphic functions in Sasakian manifolds.
method Analyzing Sasakian manifolds with specific curvature properties.
result First step towards CR analogue of Yau uniformization conjecture.
The paper confirms a specific type of Sasakian manifold's structure.
problem Characterizing Sasakian manifolds with nonnegative transverse bisectional curvature.
method Analyzing the Sasakian analogue of Yau's uniformization conjecture.
result 5-dimensional Sasakian manifolds with positive transverse bisectional curvature are CR-biholomorphic to the standard Heisenberg group.
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…
The paper proves a quadratic formality for Sasakian manifolds' representation varieties.
problem Analyzing the variety of representations of fundamental groups of Sasakian manifolds.
method Proving almost-formality of de Rham complex and vanishing cup product theorem.
result Quadratic formality of analytic germs of representation varieties.
New steady Euler flows found on 3-sphere and Sasakian manifolds.
problem Finding new steady Euler solutions on specific manifolds.
method Bifurcating from an existing ansatz and extending it to Sasakian 3-manifolds.
result Previously known solutions are not isolated, and new solutions found on 3-sphere and other Sasakian manifolds.
Given a compact, connected Lie group K, we use principal K-bundles to construct manifolds with prescribed finite-dimensional algebraic models. Conversely, let M be a compact, connected, smooth manifold which supports an almost free K-action. Under a partial formality assumption on the orbit space and a regulari…
Symplectic reduction extended to Sasakian manifolds.
problem Extending symplectic reduction to Sasakian manifolds.
method Using a semisimple group action on a Sasakian manifold, identifying the categorical quotient with the moment map zero-set quotient.
result Symplectic reduction can be applied to Sasakian manifolds.
Chern-Simons theory on a closed contact three-manifold is studied when the Lie group for gauge transformations is compact, connected and abelian. A rigorous definition of an abelian Chern-Simons partition function is derived using the Faddeev-Popov gauge fixing method. A symplectic abelian Chern-Simons partition functi…
The paper classifies Sasakian immersions into odd spheres and finds new examples.
problem Classifying Sasakian immersions into odd-dimensional spheres.
method Analyzing compact Sasakian manifolds and using classification theorems.
result Infinite families of compact Sasakian η-Einstein manifolds cannot be immersed into odd spheres. Study rigidity on CR Yamabe equation on Sasakian manifolds.
problem Proving rigidity on CR Yamabe equation on Sasakian manifolds.
method Using Jerison-Lee's differential identity and integral estimates.
result Prove that the manifold is CR isometric to Heisenberg group \(\mathbb{H}^n\).
The paper studies CR Yamabe solutions on Sasakian manifolds with nonnegative curvature.
problem Characterizing CR Yamabe solutions on Sasakian manifolds with nonnegative curvature.
method Analyzes solutions to the CR Yamabe equation in noncompact (2n+1)-dimensional Sasakian manifolds with nonnegative curvature. result The Heisenberg group H1 is the only (complete) Sasakian space with nonnegative Tanaka-Webster scalar curvature admitting a (nontrivial) positive solution.