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 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.
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.
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 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.
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 nonnegativity of the CR Paneitz operator plays a crucial role in three-dimensional CR geometry. In this paper, we prove this nonnegativity for embeddable CR manifolds. This result and previous works give an affirmative solution of the CR Yamabe problem for embeddable CR manifolds. We also show the existence of a co…
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…
In this note, we mainly focus on the existence of pseudo-Einstein contact forms, an upper bound eigenvalue estimate for the CR Paneitz operator and its applications to the uniformization theorem for Sasakian space form in an embeddable closed strictly pseudoconvex CR 3-manifold. Firstly, the existence of pseudo-Einstei…
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…
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 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 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.
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 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. 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 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.
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
A proof of the monotonicity of an entropy like energy for the heat equation on a quaternionic contact and CR manifolds is proven
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…
We study the geometry of almost contact pseudo-metric manifolds in terms of tensor fields h:=21£ξφ and ℓ:=R(⋅,ξ)ξ, emphasizing analogies and differences with respect to the contact metric case. Certain identities involving ξ-sectional curvatures are obtained. We establish necessary and su…
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…
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.
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.
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 …
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.
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.
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.
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…
In this note, we affirm the partial answer to the long open Conjecture which states that any closed embeddable strictly pseudoconvex CR 3-manifold admits a contact form θ with the vanishing CR Q-curvature. More precisely, we deform the contact form according to an CR analogue of Q%-curvature flow in a closed st…
In this paper, we study warped products of contact skew-CR submanifolds, called contact skew CR-warped products. We establish an inequality for the squared norm of the second fundamental form in terms of the warping function and the slant angle. The equality case in the statement of the inequality is investigated and s…
New operators for Q-curvature on 5D pseudohermitian manifolds.
problem Characterizing CR manifolds with Q-flat contact forms. method Constructing Q-curvature operators on specific forms. result New formula for scalar Q-curvature and cohomological characterization of CR manifolds. We propose a definition for analytic torsion of the contact complex on contact manifolds. We show it coincides with Ray-Singer torsion on any 3-dimensional CR Seifert manifold equipped with a unitary representation. In this particular case we compute it and relate it to dynamical properties of the Reeb flow. In fact th…
We construct several natural connections and Dirac type operators on a general metric contact manifold which are more sensitive to the geometric background. In the special case of CR manifolds these connections are also compatible with the CR structure and include among them the Webster connection. We also describe sev…
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…
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…
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.
R-circles in general three dimensional CR manifolds (of contact type) are the analogues to traces of Lagrangian totally geodesic planes on the sphere viewed as the boundary of two dimensional complex hyperbolic space. They form a family of certain legendrian curves on the manifold. We prove that a diffeomorphism betwee…
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…
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. We deform the contact form by the amount of the Tanaka-Webster curvature on a closed spherical CR three-manifold. We show that if a contact form evolves with free torsion and positive Tanaka-Webster curvature as initial data, then a certain Harnack inequality for the Tanaka-Webster curvature holds.
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.
New CR invariant treatment of Rumin complex via differential forms.
problem CR invariant treatment of Rumin complex.
method New treatment via differential forms, identification of balanced A∞-structures, Hodge decomposition theorems. result Sharp upper bound on Kohn--Rossi groups and CR analogue of Frölicher inequalities.
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.
We propose a global invariant σc for contact manifolds which admit a strictly pseudoconvex CR structure, analogous to the Yamabe invariant σ. We prove that this invariant is non-decreasing under handle attaching and under connected sum. We then give a lower bound on σc in a particular case.
Study on lightlike submanifolds in statistical manifold geometry.
problem Characterizing contact CR and SCR-lightlike submanifolds.
method Developed characterization theorems on integrability and geodesicity.
result Obtained results on geometry of contact CR and SCR-lightlike submanifolds.
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.
In this paper, we study contact forms on the three- dimensional Heisenberg manifold with its standard CR structure. We discover that the Q′-curvature, introduced by Branson, Fontana and Morpurgo [BFM13] on the CR three-sphere and then generalized to any pseudo-Einstein CR three manifold by Case and Yang [CY95], contr…