3-Sasaki structures linked to projective geometry.
problem Understanding 3-Sasaki structures via projective geometry.
method Establishing a connection between 3-Sasaki structures and projective structures with specific holonomy reductions.
result 3-Sasaki structures are described as projective structures with a particular holonomy reduction to the unitary quaternionic group.
Study on deformations of Einstein and nearly G2 structures in 3-Sasaki manifolds.
problem Deformation theory of Einstein and nearly G2 structures in 3-Sasaki manifolds.
method Systematic study of deformation theory, focusing on infinitesimal deformations and their parametrization via eigenfunctions of the basic Laplacian.
result Infinitesimal Einstein deformations of g 1 / 5 g_{1/\sqrt{5}} g 1/ 5 coincide with infinitesimal G 2 G_2 G 2 deformations of φ 1 / 5 \varphi_{1/\sqrt{5}} φ 1/ 5 . We give a correspondence between toric 3-Sasaki 7-manifolds S and certain toric Sasaki-Einstein 5-manifolds M. These 5-manifolds are all diffeomorphic to k#(S^2\times S^3), where k=2b_2(S)+1, and are given by a pencil of Sasaki embeddings of M in S and are given concretely by the zero set of a component of the 3-Sasaki…
A series of examples of toric Sasaki-Einstein 5-manifolds is constructed. These are submanifolds of toric 3-Sasaki 7-manifolds and such a Sasaki-Einstein 5-manifold corresponds uniquely to a toric 3-Sasaki 7-manifold. This produces examples of quasi-regular Sasaki-Einstein structures on every #k(S^2 xS^3), for k odd. T…
We study positive definite quaternionic contact ( 4 n + 3 ) (4n+3) ( 4 n + 3 ) -manifolds ( q c qc q c -manifold for short). Just like the C R CR C R -structure contains the class of Sasaki manifolds, the q c qc q c -structure admits a class of 3 3 3 -Sasaki manifolds with integrable distribution isomorphic to s u ( 2 ) \mathfrak{su}(2) su ( 2 ) . A big difference concerning the inte…
New eigenvalue bounds for 3-Sasaki metrics improve previous estimates.
problem Estimating eigenvalues for 3-Sasaki metrics.
method Improved Lichnerowicz-Obata type estimates for scalar sub-Laplacian.
result Lower bounds for the first non-zero eigenvalue of 3-Sasaki metrics.
Smooth SE structures on Sasaki-joins and Bott orbifolds constructed.
problem Constructing smooth Sasaki-Einstein structures on Sasaki-joins.
method Categorical relationship between Sasaki-joins and Bott orbifolds, explicit construction.
result Infinitely many smooth SE structures up to dimension eleven, conjecture for all odd dimensions.
Classifies invariant spinors on nine sphere types.
problem Classifying invariant spinors on homogeneous spheres.
method Using spin representation and exterior forms, classify and describe invariant spinors.
result Determine dimensions and explicit descriptions of invariant spinors.
Study curvature properties of a specific type of Sasaki manifolds.
problem Investigate curvature of 3- ( α , δ ) (α,δ) ( α , δ ) -Sasaki manifolds. method Analyze both Riemannian and canonical connection curvatures, focusing on curvature operators and their spectra.
result Describe the spectrum of curvature operators and find distinguished eigenforms.
We consider the stability of Sasaki-extremal metrics under deformations of the complex structure on the Reeb foliation. Given such a deformation preserving the action of a compact subgroup of the automorphism group of a Sasaki-extremal structure, a sufficient condition is given involving the nondegeneracy of the relati…
Study variational problem on manifold with special distributions.
problem Generalize Einstein metrics on manifold with multiple distributions.
method Define functional of pseudo-Riemannian metric and contorsion tensor, prove critical pairs make distributions totally umbilical.
result Metrics in critical pairs make all distributions totally umbilical.
In this note, we describe the geometry of the quaternionic Heisenberg groups from a Riemannian viewpoint. We show, in all dimensions, that they carry an almost 3 3 3 -contact metric structure which allows us to define the metric connection that equips these groups with the structure of a naturally reductive homogeneous sp…
Formula derived for Laplace-Beltrami spectrum on homogeneous spaces.
problem Calculating the spectrum of the Laplace-Beltrami operator on homogeneous spaces.
method Formula derivation based on eigenvalues of a generalized Casimir operator and spherical representations.
result First detailed computation and investigation of the spectrum for a family of metrics on the Aloff-Wallach manifold.
In the first part, we define and investigate new classes of almost 3-contact metric manifolds, with two guiding ideas in mind: first, what geometric objects are best suited for capturing the key properties of almost 3-contact metric manifolds, and second, the newly defined classes should admit 'good' metric connections…
Extends Tian theorem to Vaisman manifolds for approximations.
problem Approximating Vaisman metrics by immersions/embeddings.
method Study Vaisman metrics on compact manifolds.
result Extend Tian's theorem to Vaisman manifolds.
In 1972, K. Kenmotsu studied a class of almost contact Riemannian manifolds. Later, such a manifold was called a Kenmotsu manifold. This paper, we studied Kenmotsu manifolds with ( 2 n + s ) (2n+s) ( 2 n + s ) -dimensional s − s- s − contact metric manifold and this manifold, we have called generalized Kenmotsu manifolds. Necessary and sufficient c…
Study on a new type of manifolds that generalize almost C-manifolds.
problem Understanding weak nearly C-manifolds and their properties.
method Analyzing conditions for local Riemannian product structures and characterizing specific dimensions.
result Conditions for a weak nearly C-manifold to become locally a Riemannian product and characterization of specific dimensions.
Stabilized convex symplectic manifolds are equivalent to flexible Weinstein manifolds.
problem Understanding the equivalence between stabilized convex symplectic manifolds and flexible Weinstein manifolds.
method Analyzing the homotopy type and symplectic properties of the manifolds.
result Stabilized convex symplectic manifolds are symplectomorphic to flexible Weinstein manifolds.
New class of complex manifolds defined, properties studied.
problem Understanding properties of complex manifolds.
method Introduced and studied wHHR manifolds, proved metric equivalence.
result Bergman and Kobayashi metrics are biLipschitz equivalent for wHHR Stein manifolds.
A selfsimiar manifold is a Riemannian manifold ( M , g ) \left(M,g\right) ( M , g ) endowed with a homothetic vector field ξ ξ ξ . We characterize global selfsimilar manifolds and describe the structure of local selfsimilar manifolds. We prove that any selfsimilar manifold with a potential homothetic vector field is a conical Riemannian ma…
Smoothly embed 3-manifolds in 5-manifolds, simplifying topological to smooth.
problem Embedding 3-manifolds smoothly in 5-manifolds.
method Homotopy and small homotopy to achieve smooth embeddings.
result Locally flat embeddings are homotopic to smooth ones.
The study provides a structure theorem for a new class of noncompact 3-manifolds.
problem Understanding a new class of noncompact 3-manifolds.
method Proved a structure theorem for irreducible open graph manifolds.
result A canonical 'reduced' decomposition of irreducible open graph manifolds along embedded, incompressible 2-tori.
Condition for intersection of real flag manifolds in complex flag manifold.
problem Intersection conditions of real flag manifolds in a complex flag manifold.
method Condition given in terms of symmetric triad, antipodal intersection proven.
result Intersection of real flag manifolds is antipodal.
The paper studies extended quasi-Einstein manifolds with special geometric properties and solitons.
problem Exploring new types of manifolds in general relativity.
method Generalization of existing manifolds and construction of specific examples.
result Existence and properties of extended quasi-Einstein manifolds with solitons.
Conditions for flat manifolds as cusp cross-sections in arithmetic hyperbolic manifolds.
problem Determining when a flat manifold can be a cusp cross-section in arithmetic hyperbolic manifolds.
method Analyzing rational representations of holonomy groups and quasi-arithmetic manifolds.
result Conditions for a flat manifold to appear as a cusp cross-section in every commensurability class of arithmetic hyperbolic manifolds.
Classifies 3-manifolds from simplified (2,0)-trisections of 4-manifolds.
problem Classifying 3-manifolds from simplified (2,0)-trisections of 4-manifolds.
method Classifies vertical 3-manifolds as preimages of arcs on the plane for simplified (2,0)-trisection maps.
result Each 6-tuple of vertical 3-manifolds determines the source 4-manifold uniquely up to orientation reversing diffeomorphisms.
Study on 3D manifolds with specific tensor structures and their properties.
problem Characterizing 3D Riemannian manifolds with tensor structures.
method Investigation of locally conformal Riemannian product manifolds and their associated structures.
result Conditions for additional structures to be parallel and properties of almost Einstein and Einstein manifolds.
New manifold type PNDP-manifold defined with Einstein warped product structure.
problem Defining manifolds with non-standard dimensions.
method Einstein warped product manifold with special base and fiber structures.
result PNDP-manifolds are Einstein warped product manifolds with specific base and fiber properties.
Optimal inequalities for systole, inradius, and volume in hyperbolic 3-manifolds
problem Establishing optimal inequalities relating systole, inradius, and volume in hyperbolic 3-manifolds
method Using systole-volume inequalities for extremal manifolds
result Extremal manifolds for systole, inradius, and volume are identified
The paper explores F-manifolds and metrics, constructing canonical structures.
problem Understanding relationships between F-manifolds and metrics.
method Construction of canonical flat F-manifolds and homogeneous Riemannian F-manifolds.
result Construction of a canonical flat F-manifold associated to an arbitrary Riemannian F-manifold.
Defines s-manifolds and s-manifolds with corners for symplectic applications.
problem Creating a framework for singular manifolds with useful properties.
method Introducing categories of stratified manifolds and manifolds with corners.
result Fundamental classes and transverse fibre products in s-manifolds.
Embeds 3-manifolds in symplectic 4-manifolds with constraints.
problem Embedding 3-manifolds in symplectic 4-manifolds with topological and smooth properties.
method Topological and smooth embeddings, using homology cobordism and obstructions.
result 3-manifolds can be embedded in symplectic 4-manifolds with specific conditions.
We introduce a new general class of metric f-manifolds which we call (nearly) trans-S-manifolds and includes S- manifolds, C-manifolds, s-th Sasakian manifolds and generalized Kenmotsu manifold studied previously. We prove their main properties and we present many examples which justify their study.
The study provides homological characterizations for Q Q Q -manifolds and l 2 l_2 l 2 -manifolds.
problem Density of maps in characterizing Q Q Q -manifolds and l 2 l_2 l 2 -manifolds. method Investigates weakening the density of Z n Z_n Z n -maps and Z Z Z -maps to homological maps. result Obtains homological characterizations for Q Q Q -manifolds and l 2 l_2 l 2 -manifolds. A locally conformally Kähler (LCK) manifold M M M is one which is covered by a Kähler manifold M ~ \tilde M M ~ with the deck transform group acting conformally on M ~ \tilde M M ~ . If M M M admits a holomorphic flow, acting on M ~ \tilde M M ~ conformally, it is called a Vaisman manifold. Neither the class of LCK manifolds nor that of Vais…
This paper provides a new method to construct b b b -symplectic toric manifolds from toric manifolds.
problem Classifying and constructing b b b -symplectic toric manifolds. method A new method to construct b b b -symplectic toric manifolds from toric manifolds. result This new method allows for the decomposition of b b b -symplectic toric manifolds into toric manifolds. 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.
The study classifies Kähler-Frobenius manifolds and their properties.
problem Classifying Kähler-Frobenius manifolds and understanding their structure.
method Using Topological Quantum Field Theory and Frobenius manifold theory.
result All flat compact Kähler manifolds are Frobenius manifolds and are classified.
Products of LCK manifolds do not admit LCK structures.
problem Whether products of compact complex manifolds admit LCK metrics.
method Classifying known LCK manifolds and proving non-existence of LCK structures in product cases.
result Products of LCK manifolds do not admit LCK structures.
Study geodesics on infinite-dimensional manifolds using Finsler structures.
problem Geodesics on Fréchet manifolds of Riemannian metrics.
method Establish Riemann-Finsler structures, prove existence and minimality of geodesics, derive Euler-Lagrange equations.
result Geodesics on Fréchet manifolds of Riemannian metrics are length minimizing and satisfy Euler-Lagrange equations.
New proof shows compact homogeneous LCK manifolds are Vaisman.
problem Proving compact homogeneous LCK manifolds are Vaisman.
method Using homogeneous LCK manifolds with potential and a new metric construction.
result Compact homogeneous LCK manifolds are Vaisman.
Paper proves existence of minimal surfaces in hyperbolic 3-manifolds.
problem Existence of minimal surfaces in non-compact 3-manifolds.
method Analyzes specific cases of hyperbolic 3-manifolds with bounded geometry and rank-1 cusps.
result Proves existence of minimal surfaces in various hyperbolic 3-manifolds.
The paper proves Liouville theorems for holomorphic maps on pseudo-Hermitian manifolds.
problem Proving Liouville theorems for holomorphic maps on pseudo-Hermitian manifolds.
method Analyzing maps between different classes of pseudo-Hermitian manifolds, using curvature assumptions.
result Holomorphic maps are constant under certain curvature conditions.
A universal branched 3-manifold W W W characterizes Sol 3-manifolds.
problem Characterizing Sol 3-manifolds
method Using a universal branched 3-manifold W W W and a regular language result A closed 3-manifold M M M immerses into W W W if and only if M M M admits a Sol structure. Proves almost flat spin^c manifolds bound compact manifolds.
problem Proving almost flat spin^c manifolds bound compact manifolds.
method Long-standing conjecture of Farrell--Zdravkovska and S. T. Yau settled.
result Every almost flat spin^c manifold bounds a compact orientable manifold.
The paper characterizes graph manifolds using fold maps and embeddability of polyhedra.
problem Understanding the global topologies of graph manifolds.
method Using fold maps into the plane and embeddability of polyhedra in 3-manifolds.
result Characterizes graph manifolds via fold maps and polyhedra embeddability.
The paper defines quasi-isometry for almost contact metric manifolds and explores its properties.
problem Understanding quasi-isometry in almost contact metric manifolds.
method Definition and study of quasi-isometry for almost contact metric manifolds.
result Established a relation between scalar curvature and quasi-isometric constants.
The study identifies flat manifolds with unique cusp cross-sections in arithmetic hyperbolic manifolds.
problem Characterizing flat manifolds that have unique cusp cross-sections in arithmetic hyperbolic manifolds.
method Algebraic characterization of cusp cross-sections in arithmetic hyperbolic manifolds.
result Construction of flat manifolds with unique cusp cross-sections and proof of their existence in all dimensions n ≥ 32 n \geq 32 n ≥ 32 .