The paper explores properties of CRS bi-warped product submanifolds in Sasakian and cosymplectic manifolds.
problem Characterizing CRS bi-warped product submanifolds in Sasakian and cosymplectic manifolds. method Analyzing the geometric properties and inequalities of CRS bi-warped product submanifolds in Sasakian and cosymplectic manifolds. result Proved the existence and properties of CRS bi-warped product submanifolds in both types of manifolds. Study properties of bi-warped product submanifolds in specific geometric spaces.
problem Characterize geometric properties of bi-warped product submanifolds.
method Analyze squared norm of second fundamental form and warping functions.
result Relationship between squared norm and warping functions for proper slant submanifolds.
Study on submanifolds in nearly Kaehler manifolds with new inequality.
problem Characterizing submanifolds in nearly Kaehler manifolds.
method Analyzing bi-warped product submanifolds and proving an inequality.
result Proved a sharp inequality for submanifolds in nearly Kaehler manifolds.
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 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 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.
CR and conformal maps coincide on certain stratified groups.
problem Understanding mappings on stratified groups with CR structures.
method Analyzing conformal, CR, and anti-CR maps on specific Lie groups.
result Conformal maps coincide with CR and anti-CR maps on these groups.
The paper studies CR geometry and introduces a new quotient method.
problem CR geometry in arbitrary codimension.
method Levi-Kahler quotient for constructing Kahler metrics from CR structures.
result Explicit descriptions and characterizations of Levi-Kahler quotients of toric CR manifolds, especially products of odd dimensional spheres.
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…
The article surveys recent results on warped and CR-warped product submanifolds in Kaehler manifolds.
problem Characterizing submanifolds in Kaehler manifolds using warped and CR-warped products.
method Analyzing properties of warped and CR-warped products in the context of Kaehler manifolds.
result Presented several results on the geometry of warped and CR-warped product submanifolds.
Study CR submanifolds of nearly Kähler S³ × S³ using properties of its almost product structure.
problem Characterize CR submanifolds of nearly Kähler S³ × S³.
method Analyze CR submanifolds in relation to the canonical almost product structure on vector bundles.
result Identify key properties of CR submanifolds related to the almost product structure.
Paper shows CR Q-curvature orthogonal to CR pluriharmonic functions.
problem Understanding CR Q-curvature and pluriharmonic functions on CR manifolds. method Obtained a cohomological expression for the integral of CR Q-curvature and pluriharmonic functions. result CR Q-curvature is orthogonal to CR pluriharmonic functions. The purpose of this paper is to study the canonical foliations of a quaternion CR-submanifold of an almost quaternion Kaehler product manifold.
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 paper proves a new inequality for CR-warped product submanifolds in complex space forms.
problem Proving a new inequality for CR-warped product submanifolds in complex space forms.
method Developed a first Chen inequality for CR-warped product submanifolds in complex space forms.
result The bound is sharp and uniform in the sign of the holomorphic sectional curvature.
The paper explores unique properties of Kähler manifolds without shared CR-submanifolds.
problem Characterizing Kähler manifolds with specific CR-submanifolds.
method Analyzing properties of Kähler and CR-submanifolds in complex manifolds.
result Kähler manifolds without shared CR-submanifolds have distinct properties.
Paper establishes a general inequality for warped product CR-submanifolds in Kähler manifolds.
problem Finding a general inequality for warped product contact CR-submanifolds in Kähler manifolds.
method Using the Gauss equation, the paper establishes a general inequality for warped product contact CR-submanifolds in Sasakian and Kenmotsu manifolds.
result The paper proves an optimal general inequality for warped product contact CR-submanifolds in both Sasakian and Kenmotsu manifolds.
The paper characterizes warped product submanifolds in LCS-manifolds.
problem Characterizing warped product submanifolds in LCS-manifolds.
method Analyzing contact CR-warped product and pseudo-slant submanifolds, and constructing examples.
result There do not exist any proper warped product bi-slant submanifolds of LCS-manifolds.
The paper studies geometric properties of a specific type of submanifolds in Kaehler manifolds.
problem Analyzing the geometry of pointwise semi-slant warped products in locally conformal Kaehler manifolds.
method The study extends Chen's inequality for CR-warped product submanifolds and investigates equality cases.
result Several results extend Chen's inequality for CR-warped product submanifolds in Kaehler manifolds.
Study new types of submanifolds in Kenmotsu manifolds.
problem Characterize and study new types of warped product skew CR-submanifolds.
method Interchange factors of warped product skew CR-submanifolds and derive lower bounds of the square norm of second fundamental form.
result Derived a lower bound of the square norm of second fundamental form with equality cases.
Recently, M. Atcken studied Contact CR-warped prod- uct submanifolds in cosymplectic space forms and established gen- eral sharp inequalities for CR-warped products in a cosymplectic manifold [1]. In the present paper, we obtain an inequality for the squared norm of the second fundamental form in terms of constant ?φ…
We describe the automorphisms of a singular multicontact structure, that is a generalisation of the Martinet distribution. Such a structure is interpreted as a para-CR structure on a hypersurface M of a direct product space R^2 x R^2. We introduce the notion of a finite type singularity analogous to CR geometry and, al…
The CR δ-invariant for CR-submanifolds was introduced in a recent article [B. Y. Chen, An optimal inequality for CR-warped products in complex space forms involving CR δ-invariant, Internat. J. Math. 23} (2012), no. 3, 1250045 (17 pages)]. In this paper, we prove two new optimal inequalities for anti-holomorphic su…
Study on warped product pointwise semi-slant submanifolds in Sasakian and cosymplectic manifolds.
problem Investigate the geometry of pointwise semi-slant submanifolds and their warped products.
method Using the notion of pointwise slant submanifolds, investigate the geometry of pointwise semi-slant submanifolds and their warped products in Sasakian and cosymplectic manifolds.
result Prove the existence of no proper pointwise semi-slant warped product submanifolds other than contact CR-warped products in Sasakian manifolds.
Study CR-statistical submanifolds in holomorphic statistical spaces.
problem Characterize CR-statistical submanifolds and their properties.
method Optimization technique to relate Ricci curvature and mean curvature.
result Established relationship between Ricci curvature and mean curvature.
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.
n this paper, we obtain a geometric inequality between the length of the second fundamental form and the length of Lee form in terms of the warping function for a CR-warped product submanifold in a locally conformal Kaehler space form. The equality case is also investigated. Furthermore, the inequality is discussed for…
This is the first of two papers, in which we prove some properties of the Webster scalar curvature flow. More precisely, we establish the long-time existence, L^p convergence and the blow-up analysis for the solution of the flow. As a by-product, we prove the convergence of the CR Yamabe flow on the CR sphere. The resu…
The paper derives inequalities for contact CR-warped product submanifolds in cosymplectic space forms.
problem Establishing inequalities for contact CR-warped product submanifolds in cosymplectic space forms.
method Using the Gauss equation and hypotheses for cosymplectic and nearly cosymplectic manifolds, the paper derives inequalities for the norm of the second fundamental form and the shape operator.
result The contact warped product submanifolds in cosymplectic manifolds exhibit a geometric property called D1-minimality, leading to an optimal general inequality. Unified CR-twistor spaces for G2 and Spin(7) structures.
problem Formally integrable CR-structures on manifolds with G2 or Spin(7) structures. method Generalized LeBrun's, Rossi's, and Verbitsky's construction to CR-twistor spaces for manifolds with VCP structures.
result Torsion tensor properties on CR-twistor spaces for G2 and Spin(7) structures. Study shows nonvanishing CR curvature on Grauert tube boundaries.
problem Analyzing CR curvature on Grauert tube boundaries.
method Two recent formulas for Cartan CR-curvature of local smooth hypersurfaces in C^2.
result Nonvanishing Cartan CR-curvature on Grauert tube boundaries.
CR-UOT improves matching of heterogeneous single-cell omics profiles.
problem Matching nonnegative finite Radon measures across heterogeneous spaces.
method Cost-regularized unbalanced optimal transport (CR-UOT) framework.
result CR-UOT improves alignment of heterogeneous single-cell omics profiles.
We characterise the integrability of any co-CR quaternionic structure in terms of the curvature and a generalized torsion of the connection. Also, we apply this result to obtain, for example, the following. (1) New co-CR quaternionic structures built on vector bundles over a quaternionic manifold M, whose twistor space…
We study the heat kernel of the sub-Laplacian L on the CR sphere S2n+1. An explicit and geometrically meaningful formula for the heat kernel is obtained. As a by-product we recover in a simple way the Green function of the conformal sub- Laplacian -L + n2 that was obtained by Geller [12], and also get an explicit formu…
Study reveals hidden accidental parabolics in complex hyperbolic geometry.
problem Understanding hidden parabolics in complex hyperbolic geometry.
method New technique to show ideal boundary of Ford domain is an infinite-genus handlebody.
result 3-manifold at infinity of Δ4,∞,∞;∞ is the complement of chain link 814. Study CR Yamabe constant and CR structures on manifolds.
problem Understanding CR Yamabe constant and its role in CR geometry.
method Developed integral formulae and constructed families of CR structures.
result Found an infinite family of CR structures with varying CR Yamabe constants.
Paper explores existence and nonexistence of submanifolds in contact geometry.
problem Existence and nonexistence of warped product submanifolds in almost contact manifolds.
method Generalizes results for existence and nonexistence, constructs examples, addresses open problems.
result Offers decisive results for existence and nonexistence of submanifolds.
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 CR Yamabe constant, flow, and soliton on CR manifolds.
problem Analyzing CR Yamabe constant, flow, and soliton on CR manifolds.
method Using maximum principles, proving uniqueness for CR Yamabe flow, and studying properties of CR Yamabe soliton.
result Uniqueness theorem for CR Yamabe flow and properties of CR Yamabe soliton.
Connected sum of CR manifolds with positive CR Yamabe constant is possible.
problem Establishing the existence of a CR structure with positive CR Yamabe constant for connected sums of CR manifolds.
method Analyzing the properties of connected sums of CR manifolds with positive CR Yamabe constant.
result The connected sum of M1 and M2 admits a CR structure with positive CR Yamabe constant. Connected sum of CR manifolds preserves positive CR Yamabe constant.
problem Understanding CR structures on connected sums of spherical CR manifolds.
method Analyzing spherical CR manifolds with positive CR Yamabe constant.
result Connected sum of CR manifolds with positive CR Yamabe constant also admits a spherical CR structure with positive CR Yamabe constant.
Solves CR Poincaré-Lelong equation on CR manifolds, revealing structures and solitons.
problem Solving CR Poincaré-Lelong equation on CR manifolds.
method Solves CR Poisson equation on CR (2n+1)-manifolds with specific curvature properties. result Discovers structures and CR Yamabe steady solitons on complete noncompact Sasakian manifolds.
CR Yamabe flow converges on specific CR manifolds.
problem Convergence of CR Yamabe flow on compact CR manifolds.
method Proving convergence for n=1 or spherical manifolds. result CR Yamabe flow converges under specific conditions.
B. Y. Chen established sharp inequalities between certain Riemannian invariants and the squared mean curvature for submanifolds in real space form as well as in complex space form. In this paper we generalize Chen inequalities for submanifolds of Bochner Kaehler manifolds. Moreover, we consider CR-warped product subman…
Counterexamples found for CR manifolds with mixed signature.
problem Finding counterexamples to the Lichnerowicz conjecture for CR manifolds with mixed signature.
method Constructing specific CR manifolds with mixed signature properties.
result Found counterexamples to the Lichnerowicz conjecture for CR manifolds with mixed signature.
We construct a Hopf action, with an invariant trace, of a bicrossed product Hopf algebra $\cH=\big( \cU(\Fg_1) \acr \cR(G_2) \big)^{\cop}$ constructed from a matched pair of Lie groups G1 and G2, on a convolution algebra $\cA=C_c^{\ify}(G_1)\rtimes G_2^δ$. We give an explicit way to construct Hopf cyclic cohomolo…
CR structure on S³ with non-compact solutions to CR Yamabe problem.
problem Existence of non-compact solutions to CR Yamabe problem.
method Deforming standard CR structure of S³, using Lyapunov-Schmidt method.
result Existence of a blowing-up sequence of solutions.
CR-harmonic maps defined for pseudoconvex manifolds.
problem Defining CR-harmonic maps in CR geometry.
method Developing renormalized energy and CR covariant subelliptic PDE.
result CR-harmonic maps satisfy a CR covariant subelliptic PDE.