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.
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.
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.
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…
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.
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.
Let Q→M be a principal G-bundle, and B0 a connection on Q. We introduce an infinitesimal homogeneity condition for sections in an associated vector bundle Q×GV with respect to B0, and, inspired by the well known Ambrose-Singer theorem, we prove the existence of a connection which satisfies a syst…
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.
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.
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…
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 …
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.
The main result of this article provides a characterization of reductive homogeneous spaces equipped with some geometric structure (non necessarily pseudo-Riemannian) in terms of the existence of certain connection. The result generalizes the well-known result of Ambrose and Singer for Riemannian homogeneous spaces, as…
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…
We introduce horizontal holonomy groups, which are groups defined using parallel transport only along curves tangent to a given subbundle D of the tangent bundle. We provide explicit means of computing these holonomy groups by deriving analogues of Ambrose-Singer's and Ozeki's theorems. We then give necessary and suf…
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.
We provide a new perspective on parallel 2-transport and principal 2-group bundles with 2-connection. We define parallel 2-transport as a 2-functor from the thin fundamental 2-groupoid to the 2-category of 2-group torsors. The definition of the 2-category of 2-group torsors is new, and we develop the tools necessary fo…
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…
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.
New connections found in higher-dimensional geometries with skew-torsion.
problem Finding metric connections with skew-torsion on metric f-manifolds. method Analyzing properties of Reeb vector fields and Nijenhuis tensor to construct a unique connection.
result A natural generalization of adapted connections in higher dimensions, including parallel skew-torsion.
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.
A new connection in Finsler geometry unifies various types of connections.
problem Introducing a unified connection in Finsler geometry.
method Using the pullback formalism, a new linear connection is introduced and investigated.
result The existence and uniqueness of the new connection are proved intrinsically.
A (J2=±1)-metric manifold has an almost complex or almost product structure J and a compatible metric g. We show that there exists a canonical involution in the set of connections on such a manifold, which allows to define a projection over the set of connections adapted to J. This projection sends the Le…
We compute all the simply connected homogeneous and infinitesimally homogeneous surfaces admitting one or more invariant affine connections. We find exactly six non equivalent simply connected homogeneous surfaces admitting more than one invariant connections and four classes of simply connected homogeneous surfaces ad…
New normalization condition for sub-Riemannian connections.
problem Normalizing connections on sub-Riemannian manifolds.
method Formulated in terms of Cartan connections, depends on curvature's first degree of homogeneity.
result A compatible partial affine connection can be uniquely extended to a full affine connection and a grading of the tangent bundle.
Odd connections on supermanifolds are defined and their properties studied.
problem Defining and understanding odd quasi-connections on supermanifolds.
method Examined odd quasi-connections, defined torsion and curvature, and identified special classes.
result Odd connections on supermanifolds are shown to have torsion and curvature tensors.
The paper classifies Lorentzian Lie groups based on Codazzi tensors and quasi-statistical structures.
problem Classifying Lorentzian Lie groups based on specific tensor properties.
method Classification of three-dimensional Lorentzian Lie groups based on Ricci tensors and quasi-statistical structures associated with different affine connections.
result The paper classifies three-dimensional Lorentzian Lie groups based on Codazzi tensors and quasi-statistical structures associated with Bott, canonical, and Kobayashi-Nomizu connections.
The paper proves monotonicity formulas for minimal connections and their applications.
problem Understanding critical points of volume functionals in Riemannian geometry.
method Developed monotonicity formulas for minimal connections under specific conditions.
result Established vanishing theorems for minimal connections on Euclidean spaces and dDT connections on G2-manifolds.
The paper solves the Integration Problem for principal connections.
problem Describing discrete connections associated with a principal connection.
method Using the Lie or derivative functor to induce connections on the principal bundle.
result For flat principal connections, the Integration Problem has a unique solution among flat discrete connections.
This article is a continuation of my former article "On Connectivity Spaces". After some brief historical references relating to the subject, separation spaces and then adjoint notions of connective representation and connective foliation are developed. The connectivity order previously defined only in the finite case …
Study of multiplicative connections in Lie groupoids.
problem Defining and understanding multiplicative connections in Lie groupoids.
method Definition and study of multiplicative connections satisfying compatibility with the groupoid structure.
result Identification of the obstruction to the existence of a multiplicative connection.
In this paper, we study non integrable distributions in a Riemannian manifold with a semi-symmetric metric connection, a semi-symmetric non-metric connection and a statistical connection. We obtain the Gauss, Codazzi, and Ricci equations for non integrable distributions with respect to the semi-symmetric metric connect…
Develops torsion dual connections for statistical manifolds.
problem Defining statistical manifolds using dual connections.
method Introduces torsion dual connections and proves their properties.
result Curvature tensor of torsion dual connections has specific divergence.
In this paper, we compute canonical connections and Kobayashi-Nomizu connections and their curvature on three-dimensional Lorentzian Lie groups with some product structure. We define algebraic Ricci solitons associated to canonical connections and Kobayashi-Nomizu connections. We classify algebraic Ricci solitons assoc…
Extends connections on Lie groupoids, proving completeness conditions.
problem Existence and completeness of multiplicative connections on Lie groupoid fibrations.
method Introduces and investigates multiplicative Ehresmann connections on Lie groupoid fibrations.
result Conditions for completeness of multiplicative connections on Lie groupoid fibrations.
Study on solvable Lie groups with specific Weyl connections.
problem Characterizing solvable Lie groups with invariant stretched non-positive Weyl connections.
method Analyzing structure and classification of solvable Lie groups.
result Classification of solvable Lie groups and compact solvmanifolds with invariant SNP connections.
The study connects conic connections and torsion-free principal connections on G-structures.
problem Relating torsion tensors of principal connections to characteristic conic connections.
method Formulating and verifying conditions for the existence of characteristic conic connections implying torsion-free principal connections.
result Conditions for the existence of characteristic conic connections imply the existence of torsion-free principal connections, verified for adjoint varieties of simple Lie algebras.
Defines semi-symmetric metric connections on differential forms.
problem Analyzing connections on differential forms.
method Defined and studied semi-symmetric metric connections, computed their curvature and Ricci tensors, and analyzed Lie derivatives.
result Derived Gauss-Codazzi-Ricci equations and properties of canonical, Schouten, and Vrancreanu connections.
Sprays on Frechet manifolds connect connections and tangent structures.
problem Characterizing linear symmetric connections on Frechet manifolds.
method Constructing connection maps and linear symmetric connections on tangent and second-order tangent bundles using sprays.
result A bijective correspondence exists between linear symmetric connections on tangent bundles and sprays.
New connections found with specific torsion properties.
problem Understanding metric connections with specific torsion properties.
method Described Lorentzian manifolds with metric connections having parallel, skew-symmetric torsion.
result Found new Lorentzian manifolds with metric connections having parallel, skew-symmetric torsion.
Defines a new natural connection on Riemannian Π-manifolds.
problem Characterizing natural connections on Riemannian Π-manifolds.
method Introducing and analyzing the first natural connection with torsion.
result Relations between the first natural connection and Levi-Civita connection are established.
In this paper, we prove a local index theorem for the DeRham Hodge-laplacian which is defined by the connection compatible with metric. This connection need not be the Levi-Civita connection. When the connection is Levi-Civita connection, this is the classical local Gauss-Bonnet-Chern theorem.
This paper explores how local behavior of meromorphic connections on the projective line determines the global connection.
problem Determining the global meromorphic connection based on specified local behavior at singular points.
method Expository discussion of various problems related to meromorphic connections with specified local behavior, including Deligne-Simpson and rigidity problems.
result The existence and nonemptiness of moduli spaces of meromorphic connections with specified local behavior.
Paper extends Simons theorem to F-Yang-Mills connections for instability.
problem Tackles instability of F-Yang-Mills connections. method Extends Simons theorem to F-Yang-Mills connections using Kobayashi-Ohnita-Takeuchi's method. result Derives a sufficient condition for instability of non-flat F-Yang-Mills connections. Recently the present authors introduced a general class of Finsler connections which leads to a smart representation of connection theory in Finsler geometry and yields to a classification of Finsler connections into the three classes. Here the properties of one of these classes namely the Berwald-type connections whic…
The first examples of complete projective connections are uncovered: normal projective connections on surfaces whose geodesics are all closed and embedded are complete, as are normal projective connections induced from complete affine connections with slowly decaying positive Ricci curvature.
Flat Yang-Mills connections on pinched manifolds.
problem Stability of Yang-Mills connections on compact manifolds.
method Pinching conditions and weak stability criteria.
result No non-flat weakly stable Yang-Mills connections on δ(n)-pinched compact simply-connected Riemannian manifolds.
The study classifies holomorphic projective connections on complex threefolds.
problem Characterizing holomorphic projective connections on complex threefolds.
method Analyzing properties of holomorphic projective connections on complex projective threefolds.
result Holomorphic projective connections on complex threefolds are either flat or translation invariant on abelian threefolds.