The paper introduces a statistical version of contact CR-product for Sasakian statistical manifolds.
problem Characterizing geometric properties of contact CR-submanifolds in Sasakian statistical manifolds.
method Characterization of integrability of invariant and anti-invariant distributions, development of results on specific types of contact CR submanifolds, introduction of statistical contact CR-product.
result Introduction of a statistical version of contact CR-product for Sasakian statistical manifolds.
Study examines homology of contact CR-submanifolds in complex Euclidean space.
problem Investigating the homology of contact CR-submanifolds.
method Analyzes stable integral currents and homology groups of contact CR-submanifolds in Cm. result Nonexistence of stable integral currents and vanishing theorems for homology groups.
Study optimal inequality for contact CR-warped product submanifolds in cosymplectic space forms.
problem Optimal inequality for contact CR-warped product submanifolds in cosymplectic space forms.
method Using Gauss and Codazzi equations, prove an optimal inequality.
result Prove an optimal inequality for contact CR-warped product submanifolds.
Study CR manifolds focusing on Levi and contact-nondegeneracy.
problem Investigate CR algebras and their properties.
method Locally homogeneous case analysis, cross-marked painted root diagrams.
result Establish correspondences with CR-algebra properties and combinatorics.
Normalizes pseudo-Einstein contact forms for easier analysis.
problem Understanding pseudo-Einstein contact forms.
method Constructing intrinsic CR normal coordinates using parabolic normal coordinates.
result Normal form for pseudo-Einstein contact forms.
The CR Yamabe flow converges exponentially to a contact form with flat curvature.
problem Analyzing the CR Yamabe flow with zero invariant.
method Used CR Poincaré inequality and Gagliardo-Nirenberg type interpolation inequality.
result The flow converges exponentially to a contact form with flat pseudo-Hermitian scalar curvature.
Study of CR Yamabe problem on toric contact manifolds.
problem CR Yamabe problem on toric contact manifolds.
method Equivalence to boundary value problem for elliptic PDE.
result Many contact toric manifolds admit CR structures with distinct CR Yamabe invariants.
Study on warped products in contact skew-CR submanifolds with inequality and examples.
problem Analyzing warped products in contact skew-CR submanifolds.
method Established an inequality for the squared norm of the second fundamental form.
result Derived inequality and provided non-trivial examples.
Innovates contact structure invariant, non-decreasing under operations.
problem Contact structures and CR structures on manifolds.
method Introduced invariant σ_c, proved non-decreasing under handle attaching and connected sum.
result Lower bound on σ_c in a specific case.
Study on contact forms with constant curvature on CR manifolds.
problem Existence of non-homothetic contact forms with constant Tanaka-Webster scalar curvature.
method Analysis of universal covers and profinite completions of CR manifolds.
result Existence of infinitely many non-homothetic contact forms on compact CR manifolds.
We consider locally homogeneous CR manifolds and show that, under a condition only depending on their underlying contact structure, their CR automorphisms form a finite dimensional Lie group.
We study the action of the group of contact diffeomorphisms on CR deformations of compact three-dimensional CR manifolds. Using anisotropic function spaces and an anisotropic structure on the space of contact diffeomorphisms, we establish the existence of local transverse slices to the action of the contact diffeomorph…
Defines generalized Sasakian structures in contact geometry.
problem Understanding k-contact manifolds and CR manifolds.
method Introduced generalized Sasakian structures and proved their equivalence to Sasakian structures.
result k-contact manifolds are generalized Sasakian if and only if they are classically Sasakian.
Paper proves nonnegativity of CR Paneitz operator for embeddable CR manifolds.
problem Nonnegativity of CR Paneitz operator for embeddable CR manifolds.
method Analytical proof and geometric analysis.
result Affirmative solution to CR Yamabe problem for embeddable CR manifolds.
Study vanishing theorems for CR manifolds using contact forms and Laplacian formulas.
problem Vanishing theorems for Kohn-Rossi cohomology of spherical CR manifolds.
method Used a canonical contact form and Weitzenböck-type formulae for the Kohn Laplacian.
result Results are optimal in some cases and prove vanishing theorems.
The paper explores CR structures and their leaf spaces in semi-Riemannian manifolds.
problem Classifying CR structures and their leaf spaces.
method Using unit tangent bundles and dynamical Legendrian contact structures.
result New examples of 2-nondegenerate CR structures are provided.
We classify the normal CR structures on S3 and their automorphism groups. Together with [3], this closes the classification of normal CR structures on contact 3-manifolds. We give a criterion to compare 2 normal CR structures, and we show that the underlying contact structure is, up to homotopy, unique.
The paper proves existence and eigenvalue bounds for CR structures, leading to uniformization theorems.
problem Existence and eigenvalue estimates for CR structures.
method Analyzes pseudo-Einstein contact forms and CR Paneitz operator.
result Derives eigenvalue upper bounds and uniformization theorems for CR 3-manifolds.
Warped product CR-submanifolds in Kaehlerian manifolds were intensively studied only since 2001 after the impulse given by B.Y. Chen. Immediately after, another line of research, similar to that concerning Sasakian geometry as the odd dimensional version of Kaehlerian geometry, was developed, namely warped product cont…
Study on spherical CR manifolds with non-trivial Chern classes.
problem Understanding Chern classes in spherical CR manifolds.
method Construction and proof of constraints on Chern classes.
result Topological obstruction to spherical CR structures on contact manifolds.
Defines and proves CR invariants on five-manifolds.
problem Defines and studies CR invariants on CR five-manifolds.
method Defines global secondary CR invariants and proves their linear combination.
result Any global secondary CR invariant is a linear combination of total Q′-curvature, total I′-curvature, and a local CR invariant. The study proves CR structures on specific three-manifolds are equivalent to standard structures.
problem Proving CR structures on three-manifolds are equivalent to standard structures.
method Analyzing Yamabe constant and total Q′-curvature to deduce CR equivalence. result Closed CR three-manifolds with certain curvature properties are equivalent to standard structures.
Study on almost contact pseudo-metric manifolds with geometric identities and conditions.
problem Characterize and understand the geometry of almost contact pseudo-metric manifolds.
method Analyzing tensor fields and sectional curvatures, establishing conditions for CR structures and proving Sasakian properties.
result Established necessary and sufficient conditions for CR structures and proved Sasakian properties.
The paper sets a limit on CR manifold properties.
problem Understanding properties of CR manifolds.
method Analyzes Chern classes of specific manifolds.
result Establishes an optimal constraint on CR manifolds.
In this paper we produce several new invariants for CR and contact manifolds by looking at the noncommutative residue traces of various geometric projections. In the CR setting these operators arise from the Kohn-Rossi complex and include the Szegö projections on forms. In the contact setting they stem from the general…
The CR Frankel conjecture is proven for spherical CR manifolds.
problem Proving the CR Frankel conjecture in spherical CR manifolds.
method Criterion of pseudo-Einstein contact forms and analysis of first Kohn-Rossi cohomology group.
result CR Frankel conjecture affirmed for spherical CR manifolds.
Researchers create higher-dimensional I′-curvatures and find counterexamples to the Hirachi conjecture.
problem The Hirachi conjecture in higher CR dimensions.
method Constructing higher-dimensional I′-curvatures and analyzing their properties under contact form changes. result Total integrals of I′-curvatures depend on the choice of contact form, providing counterexamples to the Hirachi conjecture. Introduces a new CR invariant for co-oriented contact structures on closed 3-manifolds
problem Distinguishing contact structures on closed 3-manifolds
method Constructs an invariant μM(ξ) associated with a contact structure ξ and open book decomposition result Shows that the first Chern classes of two tight contact structures on the 3-torus are different
Study of spectral invariants on CR contact manifolds with circle action.
problem Analytic torsion and eta-like invariants on CR contact manifolds.
method Interpret spectral series topologically and dynamically using Reeb flow.
result Spectral series can be interpreted both topologically and dynamically.
The paper defines η-normality for contact and paracontact manifolds and explores their properties.
problem Defining and characterizing η-normality for contact and paracontact manifolds. method Using the Levi-Civita covariant derivative and Tanaka-like connections.
result Existence and uniqueness of connections on η-normal manifolds. A proof of the monotonicity of an entropy like energy for the heat equation on a quaternionic contact and CR manifolds is proven
An index formula is proposed for contact transformations between contact manifolds equipped with CR structures or with fillings by symplectic manifolds. The formula generalizes the Atiyah-Singer formula and gives a conjectured formula for the index of Fourier integral operators, as well as Epstein's relative index for …
Study on Hausdorff dimension of singular CR Yamabe problem.
problem Estimating the Hausdorff dimension of the singular set in CR geometry.
method Conformal deformation of contact form to solve Yamabe problem and estimate Hausdorff dimension.
result Estimates the Hausdorff dimension of the singular set in CR geometry.
In this paper we give a survey of the constructions in math.DG/0510061 of several new invariants for CR and contact manifolds. The latter extend previous constructions of Hirachi and Boutet de Monvel. In addition, we give simple algebro-geometric arguments proving that Hirachi's invariant vanishes on strictly pseudocon…
Paper proves existence of minimum energy solutions in 5D contact spin manifolds.
problem Finding minimum energy solutions for CR Yamabe equation in 5D contact spin manifolds.
method Spinorial approach based on a positive mass theorem.
result Existence of minimum energy solutions in 5D contact spin manifolds.
Paper proves stability of pseudo-Einstein contact form existence.
problem Stability of pseudo-Einstein contact form existence under deformations.
method Deformations of real hypersurfaces in complex manifolds.
result Existence of pseudo-Einstein contact form is preserved under deformations.
Solves Neumann problem on CR manifold boundary.
problem Neumann problem on CR manifold boundary.
method Analyzes CR Yamabe operator and contact forms.
result Solves Neumann problem and finds contact form.
The paper constructs contact forms on a sphere with constant Webster curvature and applies them to CR Yamabe problems.
problem Constructing contact forms on a sphere with constant Webster curvature.
method Explicit construction of contact forms conformal to the standard CR structure on $\Sph^{2n+1}\setminus \Sph^{2k+1}$.
result Existence of infinitely many contact structures with constant Webster curvature on $\Sph^{2n+1}\setminus \Sph^{2k+1}$.
We introduce a global Cauchy-Riemann(CR)-invariant and discuss its behavior on the moduli space of CR-structures. We argue that this study is related to the Smale conjecture in 3-topology and the problem of counting complex structures. Furthermore, we propose a contact-analogue of Ray-Singer's analytic torsion. Thi…
CR Q-curvature flow solves CR manifold curvature conjecture.
problem Proving existence and convergence of CR Q-curvature flow in CR 3-manifolds.
method Deforming a contact form according to CR Q-curvature flow.
result Existence and smooth asymptotic convergence of CR Q-curvature flow.
In this note, we prove that the CR manifold which is induced from the canonical parabolic geometry of a quaternionic contact (qc) manifold via a Fefferman-type construction is equivalent to the CR twistor space of the qc manifold defined by O. Biquard.
Study of CR-submanifolds in various Lorentzian manifolds.
problem Exploring CR-submanifolds in different Lorentzian structures.
method Analyzing properties and results of CR-submanifolds in LCS, LP-cosymplectic, S, and GKM manifolds.
result Obtained results on totally umbilical and geodesic CR-submanifolds.
Contact Riemannian manifolds, with not necessarily integrable complex structures, are the generalization of pseudohermitian manifolds in CR geometry. The Tanaka-Webster-Tanno connection on such a manifold plays the role of Tanaka-Webster connection in the pseudohermitian case. We prove the contact Riemannian version of…
New invariant distinguishes tight contact structures on 3-tori.
problem Distinguishing tight contact structures on 3-manifolds.
method Introduced a new CR invariant μM(ξ) for co-oriented contact structures on 3-manifolds. result First Chern classes of two tight contact structures on 3-torus are different.
In the present paper, we study globally framed f-manifolds in the particular setting of indefinite S-manifolds for both spacelike and timelike cases. We prove that if M=N⊥×fNT is a warped CR-submanifold such that N⊥ is φ?-anti-invariant and NT is φ?-invariant, then M is a CR-product. We…
The present paper deals with the study of pseudo parallel (in the sense of Chaki and in the sense of Deszcz) contact CR-submanifolds with respect to Levi-Civita connection as well as semisymmetric metric connection of Kenmotsu manifolds and prove that these corresponding two classes are equivalent with a certain condit…
The notions of a twistor space of a contact manifold and a contact connection on such a manifold have been introduced by L. Vezzoni as extensions of the corresponding notions in the case of a symplectic manifold. Given a contact connection on a contact manifold one can define an almost CR-structure on its twistor spa…
Local flatness theorem for paraquaternionic contact structures.
problem Local flatness of paraquaternionic contact manifolds.
method Defined paraquaternionic contact conformal curvature tensor and showed local flatness condition.
result Paraquaternionic contact conformal curvature vanishing implies local flatness.