New strategy proves Ambrose conjecture for 3-manifolds using linking curves and sutured manifolds.
problem Proving the Ambrose conjecture for generic 3-manifolds.
method Defining linking curves and sutured manifolds to prove the conjecture.
result Explicit construction of linking curves that follow the conjugate descending flow.
Study on special Hermitian manifolds with specific connection properties.
problem Compact Hermitian manifolds with a particular Chern connection.
method Proved structure theorems for manifolds with parallel torsion and curvature.
result Structure theorems for manifolds with these properties.
Study extends isometric immersions using submanifolds.
problem General extension problem for isometric immersions.
method Establishing Cartan-Ambrose-Hicks theorems based on submanifolds.
result Geometric constructions of isometric extensions.
The Ambrose-Singer theorem is extended to cohomogeneity one Riemannian manifolds.
problem Characterizing isometric actions with specific orbit properties.
method Using a linear connection with covariant equations similar to the Ambrose-Singer theorem.
result Isometric cohomogeneity one foliations described in terms of such connections.
The goal of this thesis is to study the singularities of the exponential map of Riemannian and Finsler manifolds (a concept related to caustics and catastrophes), and the object known as the cut locus (aka ridge, medial axis or skeleton), to improve existing results about its structure, to look at it in new ways, and t…
Ambrose and Singer characterized connected, simply-connected and complete homogeneous Riemannian manifolds as Riemannian manifolds admitting a metric connection such that its curvature and torsion are parallel. The aim of this paper is to extend Ambrose-Singer Theorem to the general framework of locally homogeneous pse…
The paper proves manifold diffeomorphism under certain curvature conditions.
problem Proving diffeomorphism between manifolds with specific curvature properties.
method Using radial curvatures and L1-norm comparison to establish diffeomorphism. result Closed Riemannian manifolds with single cut points are diffeomorphic under certain curvature conditions.
Study how algebraic conditions on isotropy group affect Lorentzian homogeneous space geometry.
problem Understand how algebraic conditions on isotropy group affect the geometry and curvature of Lorentzian homogeneous spaces.
method Prove that a Lorentzian locally homogeneous space is locally isometric to a plane wave if it admits an Ambrose--Singer connection with indecomposable, non-irreducible holonomy.
result Generalize existing results about Lorentzian homogeneous spaces with irreducible isotropy and prove results about Lorentzian connections with parallel torsion and 2-symmetric connections.
The paper introduces a new concept of infinitesimal homogeneity for connections on bundles and applies it to prove known theorems and derive new results.
problem Understanding and proving theorems related to connections on bundles and their properties.
method Introducing infinitesimal homogeneity for sections in associated vector bundles and proving the existence of connections satisfying parallelism conditions.
result The existence of connections satisfying parallelism conditions and the ability to classify locally homogeneous and symmetric triples.
Extends submanifold theorem to general spaces.
problem Tackles submanifolds in general ambient spaces.
method Uses development of curves in positive codimension and generalizes Cartan-Ambrose-Hicks theorem.
result Provides a geometric construction of isometric immersions.
Study on BAS manifolds with parallel torsion and curvature.
problem Characterizing and classifying BAS manifolds.
method Canonical reduction theorem, classification in homogeneous settings, construction of combined geometries.
result Classification of BAS manifolds in various settings.
Analyzes canonical reductive decomposition of extrinsic homogeneous submanifolds.
problem Understanding the reductive decomposition of extrinsic homogeneous submanifolds.
method Examines Lie subgroups and reductive decompositions of homogeneous structures.
result Establishes a connection with the Ambrose-Singer theorem and homogeneous structures.
Ambrose, Palais and Singer \cite{Ambrose} introduced the concept of second order structures on finite dimensional manifolds. Kumar and Viswanath \cite{Kumar} extended these results to the category of Banach manifolds. In the present paper all of these results are generalized to a large class of Frechet manifolds. It is…
Characterizes group connections on group bundles.
problem Understanding connections on group bundles.
method Characterizes connections as affine spaces and uses the Ambrose-Singer theorem.
result Group connections form an affine space over cocycles.
Characterizes homogeneous spaces with geometric structures using connections.
problem Characterizing homogeneous spaces with various geometric structures.
method Using connections to characterize reductive homogeneous spaces.
result Generalizes Ambrose-Singer theorem to non-Riemannian geometries.
In this paper, we start from an extension of the notion of holonomy on diffeological bundles, reformulate the notion of regular Lie group or Frölicher Lie groups, state an Ambrose-Singer theorem that enlarges the one stated in \cite{Ma2}, and conclude with a differential geometric treatment of KP hierarchy. The example…
The paper classifies Hermitian manifolds with specific connection properties.
problem Classifying Hermitian manifolds with specific connection properties.
method Algebraic consideration of holonomy systems, structure theorems, and classification theorems.
result The universal cover of such Hermitian manifolds is the product of a complex Lie group and Hermitian symmetric spaces.
This paper is concerned with the holonomy of a class of spaces which includes Landsberg spaces of Finsler geometry. The methods used are those of Lie groupoids and algebroids as developed by Mackenzie. We prove a version of the Ambrose-Singer Theorem for such spaces. The paper ends with a discussion of how the results …
The paper proves compactness theorems for specific types of tensors.
problem Proving compactness theorems for Riemannian manifolds with specific tensors.
method Using h-almost Ricci tensors and generalized quasi-Einstein tensors, the paper extends previous theorems. result Theorems are extended to cases where h has at most linear growth. In this article, we give a theorem of reduction of the structure group of a principal bundle P with regular structure group G. Then, when G is in the classes of Lie groups defined by T.Robart [13], we define the closed holonomy group of a connection as the minimal closed Lie subgroup of G for which the previous theorem…
Paper shows spectra can't distinguish naturally reductive manifolds.
problem Cannot distinguish naturally reductive manifolds using Laplace-Beltrami spectrum.
method Characterized naturally reductive 2-step nilpotent Lie groups via Ambrose-Singer's structures; constructed isospectral pairs of 9-dimensional nilmanifolds.
result Spectra of Laplace-Beltrami operator can't distinguish naturally reductive manifolds from non-naturally reductive ones.
Research covers geometry, analysis, and integration on infinite-dimensional spaces.
problem Exploring geometric and analytical structures in infinite-dimensional settings.
method Analyzes numerical schemes, Lie groups, connections, and integration theory.
result Developed new methods for integration and analysis on infinite-dimensional manifolds.
Using the notion of Levi form of a smooth distribution, we discuss the local and the global problem of existence of one horizontal section of a smooth vector bundle endowed with a horizontal distribution. The analysis will lead to the formulation of a "one-leaf" analogue of the classical Frobenius integrability theorem…
New holonomy groups help solve foliation problems.
problem Conditions for foliations to be totally geodesic or with principal bundle structure.
method Introduce and compute horizontal holonomy groups, derive analogues of Ambrose-Singer's and Ozeki's theorems.
result Explicit conditions for foliations based on horizontal holonomy groups.
The classical Wilson loop is the gauge-invariant trace of the parallel transport around a closed path with respect to a connection on a vector bundle over a smooth manifold. We build a precise mathematical model of the super Wilson loop, an extension introduced by Mason-Skinner and Caron-Huot, by endowing the objects o…
The main purpose of this article is to introduce a comprehensive, unified theory of the geometry of all connections. We show that one can study a connection via a certain, closely associated second-order differential equation. One of the most important results is our extended Ambrose-Palais-Singer correspondence. We ex…
The paper proves Laplacian comparison theorems for modified m-Bakry-Emery Ricci tensors on Riemannian manifolds.
problem Analyzing modified m-Bakry-Emery Ricci tensors on Riemannian manifolds.
method Proving Laplacian comparison theorems using modified m-Bakry-Emery Ricci tensors under m≤1.
result Optimal conditions for modified m-Bakry-Emery Ricci tensors under m≤1 are derived.
The paper introduces controllable principal connections and estimates distances between bundles and spaces.
problem Estimating distances between bundles and spaces using controllable connections.
method Combining orbit theorem, Ambrose-Singer theorem, and controllable principal connections.
result Proves convergence of metrics to normal reductive homogeneous spaces.
Classification of 3-symmetric spaces with Ricci solitons.
problem Classifying 3-symmetric spaces and their Ricci solitons.
method Detailed analysis of Riemannian 3-symmetric spaces, including explicit constructions and moduli spaces.
result Explicit construction of expanding Ricci solitons on Type III spaces and generalization to any effective Lie group representation.
Defines parallel 2-transport and 2-group bundles, proving new theorems.
problem Understanding parallel transport in higher dimensions.
method Introduces a new 2-category of 2-group torsors and defines parallel 2-transport as a 2-functor.
result Proves non-Abelian Stokes and Ambrose-Singer theorems for 2-transport.
The paper proves a Laplacian comparison theorem on weighted Riemannian manifolds and applies it to diffusion processes.
problem Analyzing diffusion processes on Riemannian manifolds with weighted metrics.
method Proving a Laplacian comparison theorem and applying it to various geometric and analytic properties of diffusion processes.
result Optimal conditions on m-Bakry-Émery Ricci tensor for various geometric and analytic properties to hold on weighted complete Riemannian manifolds. Polyhedra volume conjecture supports Stoker conjecture weakly.
problem Proving the Stoker conjecture for polyhedra.
method Using an extension of Montcouquiol and Weiss' result on polyhedra angles as local coordinates.
result Volume Conjecture for polyhedra implies a weak version of the Stoker conjecture.
Survey on two non-Kähler geometry conjectures.
problem Constant holomorphic sectional curvature and Fino-Vezzoni conjectures in non-Kähler geometry.
method Survey and discussion of historical and recent developments.
result Discussion of conjectures without new results.
The paper generalizes a surgery conjecture and proves it under the Zilber-Pink conjecture.
problem Generalizing the Cosmetic Surgery Conjecture to n-cusped hyperbolic 3-manifolds. method Proves the generalized conjecture under the assumption of the Zilber-Pink conjecture, and without assuming it for n=1 and 2. result Proves the generalized Cosmetic Surgery Conjecture for n=1 and 2 without assuming the Zilber-Pink conjecture. Numerical study confirms Brennan's conjecture for a counterexample to Thurston's K=2 conjecture.
problem Thurston's K=2 conjecture and Brennan's conjecture in planar domains. method Numerical analysis of a specific counterexample to Thurston's conjecture.
result The counterexample does not contradict Brennan's conjecture.
The non-vanishing conjecture implies the abundance conjecture in certain cases.
problem Abundance conjecture in algebraic geometry.
method Proof of the abundance conjecture under specific conditions.
result The abundance conjecture holds in dimensions ≤ 5 when κ ≥ 0 and ν ≤ 1.
Proves Gromov's conjecture and answers Stoker's polyhedron conjecture.
problem Gromov's flat corner domination conjecture and Stoker's conjecture for convex polyhedra.
method Same techniques applied to prove conjectures.
result Proves Gromov's conjecture and answers Stoker's polyhedron conjecture.
This paper gives an algebraic conjecture which is shown to be equivalent to Thurston's Geometrization Conjecture for closed, orientable 3-manifolds. It generalizes the Stallings-Jaco theorem which established a similar result for the Poincare Conjecture. The paper also gives two other algebraic conjectures; one is equi…
The Burghelea conjecture is proven for many groups, but not all, with counter-examples provided.
problem Computing the periodic cyclic homology of complex group rings.
method Analyzing groups of finite asymptotic dimension and constructing counter-examples.
result The Burghelea conjecture holds for many classes of groups but not for all, with specific counter-examples provided.
Paper discusses conjectures and proves some related inequalities.
problem Unified generalization of BW and DDVV inequalities.
method Unified discussion and proofs of conjectures.
result Proves Conjecture 2 and obtains a new upper bound for Conjecture 3.
Symmetry-breaking in three differential geometry conjectures.
problem Exploring the role of symmetry in three differential geometry conjectures.
method Examining the Carathéodory, Willmore, and Lawson Conjectures through the lens of symmetry in 3D space-forms.
result Symmetry is broken, and more general ambient metrics are considered, leading to the failure of the conjectures.
We introduce a new variant of the coarse Baum-Connes conjecture designed to tackle coarsely disconnected metric spaces called the boundary coarse Baum-Connes conjecture. We prove this conjecture for many coarsely disconnected spaces that are known to be counterexamples to the coarse Baum-Connes conjecture. In particula…
Survey on topological rigidity problems and related conjectures.
problem Topological rigidity of closed aspherical manifolds.
method Review of recent results and open problems.
result Status and implications of Farrell-Jones Conjectures.
Study proves Hecke lifting conjecture for torus knots and verifies it for any framed knots.
problem Integrality structure of framed knots' quantum invariants.
method Explicit formulas of colored HOMFLY-PT invariants of torus knots, verified in a limit form for any framed knots.
result Proves Hecke lifting conjecture for torus knots and verifies it for any framed knots.
New proof shows most thin knots satisfy Cabling Conjecture.
problem Cabling Conjecture for thin knots using Heegaard Floer homology.
method Heegaard Floer homology and immersed curves techniques.
result Almost all thin knots satisfy the Cabling Conjecture.
The study confirms a conjecture for a specific type of knot.
problem Determining the topology of essential surfaces in knot theory.
method Analyzing twisted, generalized Whitehead doubles of knots.
result Twisted, generalized Whitehead doubles satisfy the Slope and Strong Slope Conjectures.
Equivalent conjectures proved for group presentations.
problem Balanced presentations of the trivial group.
method Equivalence of cyclic and satellite versions, restrictive cancellative version, stabilizations consideration.
result Original conjecture equivalent to cyclic version.
Metric SYZ conjecture proved using non-archimedean geometry.
problem Proving the metric SYZ conjecture in large generality.
method Using non-archimedean geometry to support the conjecture.
result Metric SYZ conjecture can be proved in large generality.