The article proves a lower semicontinuity property of holonomy maps.
problem Continuity properties of holonomy maps on manifolds.
method Examined continuity properties of holonomy class and restricted holonomy class maps.
result The restricted holonomy class map is lower semicontinuous.
The paper proves a mapping from a space of holonomy varieties to Teichmüller spaces, with a non-empty discrete intersection.
problem Intersection of Poincaré holonomy varieties and their properties.
method Holomorphic mapping and branched covering proof.
result Intersection of arbitrary Poincaré holonomy varieties is a non-empty discrete set.
Study connects landslide flow to integrable systems for harmonic maps.
problem Understanding the holonomy of complex landslide flow.
method Integrable systems approach to harmonic maps into symmetric spaces.
result Holonomy of complex landslide flow derived from harmonic map holonomy.
Simplified holonomy map for ruled submanifolds in graded manifolds.
problem Characterizing deformability of ruled submanifolds in graded manifolds.
method Introducing natural coordinates and higher dimensional holonomy map for ruled submanifolds.
result Characterization of singularities and deformability criterion for ruled submanifolds.
Holonomy map is a local biholomorphism for parabolic projective structures on non-compact surfaces.
problem Holonomy map properties for parabolic projective structures on non-compact surfaces.
method Proving the holonomy map is a local biholomorphism.
result Holonomy map is a local biholomorphism for parabolic projective structures on non-compact surfaces.
In this paper we introduce a generalisation of the notion of holonomy for connections over a bundle map on a principal fibre bundle. We prove that, as in the standard theory on principal connections, the holonomy groups are Lie subgroups of the structure group of the principle fibre bundle and we also derive a straight…
Holonomy maps on colored complexes link manifold topology to group theory.
problem Linking manifold topology with group theory.
method Investigates k-multilayer complexes and their holonomy maps. result Holonomy classes of complexes can represent topological problems.
The paper generalizes a theorem and introduces a new characteristic map for foliated manifolds.
problem The challenge is to generalize Bott's vanishing theorem for foliated manifolds.
method The approach involves working with the full holonomy groupoid instead of the Morita equivalent étale groupoid, leading to novel geometric representatives of characteristic classes.
result A characteristic map encoding both primary and secondary characteristic classes is introduced.
Classifies specific types of Lorentzian manifolds with unipotent holonomy.
problem Classifying closed conformally flat Lorentzian manifolds with unipotent holonomy.
method Analyzing the geometric properties and developing maps of the manifolds.
result Identifies four geometric types and two additional types of manifolds.
The paper extends Dunfield's theorem about 3-manifold holonomy maps and volume functions.
problem The birationality of the peripheral holonomy map on character varieties.
method Generalization of Dunfield's rigidity argument using volume functions.
result A conjecture about the volume function on character varieties implies birationality.
Study continuous deformations of branched projective structures on surfaces, preserving holonomy and branch points.
problem Continuous deformations of branched projective structures on closed surfaces of genus g≥2. method Schiffer variations and analysis of canonical divisors.
result Branch points are necessarily arranged on a canonical divisor when the underlying complex structure is infinitesimally preserved.
Paper proves certain closed affine manifolds without invariant lines don't exist.
problem Proving non-existence of closed affine manifolds with invariant lines.
method Developing map, holonomy, invariant line, large open subsets, modified proof.
result Developing image cannot meet invariant line if affine holonomy acts purely by translations.
Holonomy groups of metric connections converge in a monotonic way.
problem Monotonicity of holonomy groups under convergence of metric connections.
method Proving the monotonicity of holonomy groups for sequences of metric connections converging in C0. result The holonomy group of the limit connection is contained in the holonomy group of the initial connections.
In this paper we compute explicit formulas for the holonomy map for a gerbe with connection over an orbifold. We show that the holonomy descends to a transgression map in Deligne cohomology. We prove that this recovers both the inner local systems in Ruan's theory of twisted orbifold cohomology and the local system of …
Identifies holonomy of affine surfaces via meromorphic connections.
problem Holonomy of complex affine surfaces.
method Moduli space of complex affine surfaces and meromorphic connections.
result Holonomy map is a submersion for non-finite-area translation surfaces.
In order to understand the linearization problem around a leaf of a singular foliation, we extend the familiar holonomy map from the case of regular foliations to the case of singular foliations. To this aim we introduce the notion of holonomy transformation. Unlike the regular case, holonomy transformations can not be…
This article is a follow-up of ``Holonomy and Path Structures in General Relativity and Yang-Mills Theory" by Barrett, J. W. (Int.J.Theor.Phys., vol.30, No.9, 1991). Its main goal is to provide an alternative proof of this part of the reconstruction theorem which concerns the existence of a connection. A construction o…
Classifies orbit closures of mapping class group action on affine character variety.
problem Classifying orbit closures of Mod(Σ)-action on χ(Aff(C)). method Classification up to exceptions of Mod(Σ)-action on affine character variety. result The only obstruction for a non-abelian representation to be the holonomy of a branched affine structure is to be Euclidean and not have positive volume.
The study extends calibrated geometry to smooth maps and finds energy bounds.
problem Finding energy bounds for smooth maps between Riemannian manifolds.
method Generalizing calibrated submanifolds to smooth maps and applying to energy functional.
result Lower bounds to the energy of smooth maps in homotopy classes.
Conditions for torsion-free connections with specific curvature maps are derived.
problem Finding conditions for torsion-free connections with prescribed curvature.
method Using a power series approach to derive necessary and sufficient conditions for a curvature map to arise from a torsion-free connection.
result A unique torsion-free connection is derived from a given curvature map.
New construction of isoparametric submanifolds in Hilbert spaces.
problem Constructing isoparametric submanifolds in Hilbert spaces.
method Analyzing holonomy maps and connections in Hilbert spaces.
result Each component of the inverse image of an equifocal submanifold by the holonomy map is an isoparametric submanifold.
The study provides counting estimates and equidistribution for periodic orbits of hyperbolic rational maps.
problem Counting and equidistribution of periodic orbits of hyperbolic rational maps.
method Study of dynamical zeta functions twisted by characters of S1. result Non-vanishing of zeta functions on a half plane, leading to counting and equidistribution results.
The study shows how certain geometries can be mapped to simpler structures.
problem Understanding the structure of geometries with specific properties.
method Reframing known submersion results and applying them to new contexts.
result Holonomy decompositions and holomorphic submersions are derived for specific geometries.
The paper constructs a moduli space for opers and proves its symplectic properties.
problem Understanding the moduli space of opers and its symplectic structure.
method Constructing the moduli space of marked oper structures and proving properties of the holonomy map.
result The symplectic structure on the moduli space of marked complex projective structures extends to a pre-symplectic structure on the moduli space of marked opers.
We study geometry on real gerbes in the spirit of Cheeger-Simons theory. The concepts of adaptations and holonomy forms are introduced for flat connections on real gerbes. Their relations to complex gerbes with connections are presented, as well as results in loop and map spaces.
We study the limits of holonomy representations of complex projective structures on a compact Riemann surface in the Morgan-Shalen compactification of the character variety. We show that the dual R-trees of the quadratic differentials associated to a divergent sequence of projective structures determine the Morgan-Shal…
Flat connections induced over covering maps are studied and the trivial ones among them are described. In the sequel, we deal with the resulting holonomy bundles.
New proof shows certain manifolds cannot have real projective structure.
problem Demonstrating certain manifolds do not admit real projective structures.
method Reproved classification of closed real projective manifolds with nilpotent holonomy groups, using octantizability concept.
result Proved RPn#RPn and some others do not admit real projective structures. Study mirror symmetry on manifolds with exceptional holonomy groups.
problem Construct mirrors for Spin(7) and G2 manifolds.
method Apply mirror symmetry to pairs of non-compact manifolds and use CFT analysis.
result Confirm geometric mirror constructions for Spin(7) and G2 manifolds.
New geometric proof and generalization of Chen signature theorem.
problem Chen signature theorem and its generalizations.
method Topology on loops, Fréchet-Lie group, principal bundle with connection.
result Alternative geometric proof and generalization of Chen signature theorem.
Geometrically describes homology of surface bundles over surfaces.
problem Understanding the homology of surface bundles over surfaces.
method Geometric description using holonomy map.
result Observations on the homology of surface bundles, replacing a previous theorem.
The paper explores holonomy, zeta functions, and cohomology in foliated manifolds with stratified boundaries.
problem Understanding symmetries and cohomology in foliated manifolds with stratified boundaries.
method Developed a novel formalism for the Gamma-set and defined an Ihara zeta function to encode symmetries. Investigated the relationship between holonomy and zeta functions, and analyzed how the twist map impacts cohomology.
result Conjectured a duality between holonomy fixed points and the poles of the Ihara zeta function, extending to twisted cohomology classes.
A singular foliation on a complete riemannian manifold M is said to be riemannian if each geodesic that is perpendicular at one point to a leaf remains perpendicular to every leaf it meets. We prove that the regular leaves are equifocal, i.e., the end point map of a normal foliated vector field has constant rank. This …
This paper generalizes Bismut's equivariant Chern character to the setting of abelian gerbes. In particular, associated to an abelian gerbe with connection, an equivariantly closed differential form is constructed on the space of maps of a torus into the manifold. These constructions are made explicit using a new local…
In this paper we consider two generalizations of the Skyrme model. One is a variational problem for maps from a compact three-manifold to a compact Lie group. The other is a variational problem for flat connections. We describe the path components of the configuration spaces of smooth fields for each of the variational…
Let S be a closed oriented surface of genus g≥2. Fix an arbitrary non-elementary representation ρ:π1(S)→SL2(C) and consider all marked (complex) projective structures on S with holonomy ρ. We show that their underlying conformal structures are dense in the moduli space of S.
Study shows how tangle moduli spaces relate to boundary surfaces.
problem Relating tangle moduli spaces to boundary surfaces.
method Use of composition in the Weinstein category and SU(2) holonomy perturbations.
result Holonomy perturbed SU(2) traceless flat moduli space is a Lagrangian immersion.
This paper calculates the derivative of surface holonomy for non-abelian gerbes.
problem Calculating the derivative of surface holonomy for non-abelian gerbes.
method Explicit calculation of the derivative formula for surface holonomy of squares mapped into the base manifold.
result Derivation of a formula for the derivative of surface holonomy of squares mapped into the base manifold.
In this paper we establish a one-to-one correspondence between S1-gerbes with connections, on the one hand, and their holonomies, for simply connected manifolds, or their parallel transports, in the general case, on the other hand. This result is a higher-order analogue of the familiar equivalence between bundles wi…
Introduces Weyl bundle gerbe and shows it's not D-stably isomorphic.
problem Identifying and distinguishing bundle gerbes.
method Introduces cup product bundle gerbe, defines Weyl bundle gerbe, and uses holonomy to show non-isomorphism.
result Weyl bundle gerbe and basic bundle gerbe are not D-stably isomorphic.
We investigate various structures associated with the hyperbolic Markov and homological spectra of a pseudoAnosov map φ on a surface. Each unstable eigenvalue of the action of φ on first cohomolgy yields an eigen-cocycle that is transverse and holonomy invariant to the stable foliation Fs of φ. Each …
Introduces new K-theory for loop spaces with specific holonomy bundles.
problem Developing a new K-theory framework for loop spaces with exotic twists.
method Defines exotic twisted equivariant K-theory and Chern character.
result Establishes a mapping between new K-theory and cohomology.
The classification of the holonomy algebras of Lorentzian manifolds can be reduced to the classification of irreducible subalgebras h⊂so(n) that are spanned by the images of linear maps from Rn to h satisfying an identity similar to the Bianchi one. T. Leistner fou…
In this paper we review some author's results about singular holonomy of singular riemannian foliations with sections (s.r.f.s for short) and also some results of a joint work with Toeben and a joint work with Gorodski. We stress here that the condition that the leaves are compact, used in some of these results, can be…
Study of foliations' geometric and topological structures.
problem Analyzing the geometric and topological properties of transversely affine foliations.
method Attach holonomy group and quotient stack, identify reparametrisations, classify them, and study the Kato-Nakayama space.
result Holonomy group controls the geometric part, while the Kato-Nakayama space captures the topological and dynamical aspects.
This is a survey of the theory of complex projective (CP^1) structures on compact surfaces. After some preliminary discussion and definitions, we concentrate on three main topics: (1) Using the Schwarzian derivative to parameterize the moduli space (2) Thurston's parameterization of the moduli space using grafting (3) …
The rotation angle of a rolling disc is shown to be a geometric phase related to the Hopf fibration.
problem Understanding the geometric nature of rotation angles in kinematic models.
method Using the Hopf fibration and Gauss map, the geometric phase is decomposed into dynamical and geometric components.
result The geometric phase of rotation is described as the holonomy of the Hopf fibration.
A classic theorem in the theory of connections on principal fiber bundles states that the evaluation of all holonomy functions gives enough information to characterize the bundle structure (among those sharing the same structure group and base manifold) and the connection up to a bundle equivalence map. This result and…