Defines double principal bundles with applications in geometry.
problem Understanding structures in double vector bundles.
method Definition and analysis of double principal bundles.
result Double vector bundles can be realized as associated bundles of their frame bundles.
Develops n-tuple principal bundles from Lie group actions.
problem No specific problem stated; focuses on concept development.
method Compatibility condition for Lie group action on groupoid.
result Concept of n-tuple principal bundles established.
No Einstein metrics found on certain double disk bundles.
problem Existence of Einstein metrics on specific manifolds.
method Phase space barrier argument to show non-existence.
result Proves non-existence of cohomogeneity one Einstein metrics.
Global Double Field Theory is a higher-dimensional generalization of Kaluza-Klein theory.
problem Formulating a global theory for higher-dimensional gauge fields.
method Generalizing Kaluza-Klein theory to higher principal bundles and higher gauge fields.
result Higher Kaluza-Klein geometry provides a global formulation for Double Field Theory.
We show how the double vector bundle structure of the manifold of double velocities, with its submanifolds of holonomic and semiholonomic double velocities, is mirrored by a structure of holonomic and semiholonomic subgroups in the principal prolongation of the first jet group. We use the actions of these groups to con…
Paper finds conditions for special geometric structures on certain spaces.
problem Existence of specific geometric structures on double disk bundles.
method Derives a sufficient condition involving geometric data from principal orbits.
result Sufficient condition for the existence of cohomogeneity one Einstein metrics.
Constructs a triple on an Atiyah algebroid with connection.
problem Dynamics of systems on principal bundles and Atiyah algebroids.
method Constructs a Tulczyjew triple on a principal bundle with connection, then reduces to the Atiyah algebroid.
result Dynamics of systems on principal bundles and Atiyah algebroids are discussed and applied.
The paper reviews a correspondence between Double Field Theory and bundle gerbes.
problem Exploring a geometric interpretation of Double Field Theory.
method Interpreting Double Field Theory as a field theory on the total space of bundle gerbes.
result Double Field Theory can be seen as a higher geometric field theory.
Bigerbes refine 2-gerbes, representing cohomology classes.
problem Representing degree four integral cohomology classes of a space.
method Defined in terms of bisimplicial line bundles, with symmetry properties.
result Provides natural examples, including a bigerbe associated to a principal G-bundle.
In arXiv:math/0605587, the first two authors have constructed a gauge-equivariant Morse stratification on the space of connections on a principal U(n)-bundle over a connected, closed, nonorientable surface. This space can be identified with the real locus of the space of connections on the pullback of this bundle over …
New spherical Milnor spaces for diffeological groups with geometric and topological properties.
problem Understanding higher topological structures in diffeological spaces.
method Spherical Milnor construction based on quadratic normalization.
result Provides a natural setting for studying principal bundles with Z2-twists and higher cohomology. A theory of double affine and special double affine bundles, i.e. differential manifolds with two compatible (special) affine bundle structures, is developed as an affine counterpart of the theory of double vector bundles. The motivation and basic examples come from Analytical Mechanics, where double affine bundles hav…
Study of semi-principal bundles using group actions and wreath products.
problem Understanding bundles with fibers as free G-spaces. method Defining semi-principal bundles, bases, and frame bundles; using wreath products and functors.
result Semi-principal bundles can be retracted to principal bundles, preserving parallel transport.
The study of quotient structures in multi-graded bundles, including double vector bundles.
problem Understanding quotients of multi-graded bundles, especially double vector bundles.
method Analyzing quotients as towers of affine bundles and constructing normal bundles.
result Any quotient of multi-graded bundles fits into a tower of affine bundles.
In this study, we generalize double tangent bundles to double jet bundles. We present a secondary vector bundle structure on a 1-jet of a vector bundle. We show that 1-jet of a vector bundle carries two vector bundle structures, namely primary and secondary structures. We also show that the manifold charts induced by p…
Introduces generalized principal bundles and connections, linking them to standard gauge theories.
problem Generalized principal bundles and connections in field theories.
method Local coordinate transformation laws and horizontal lifts.
result Generalized principal connections are associated to Lie group fiber bundle connections.
Elliptic Chern characters and Atiyah-Witten formula generalized to double loop spaces.
problem Generalizing classical Atiyah-Witten formula to double loop spaces.
method Constructing elliptic Chern and Bismut-Chern characters, defining elliptic holonomy, and using equivariant twisted parallel transport.
result Established elliptic Atiyah-Witten formula on double loop space.
In this paper, we construct a category of short exact sequences of vector bundles and prove that it is equivalent to the category of double vector bundles. Moreover, operations on double vector bundles can be transferred to operations on the corresponding short exact sequences. In particular, we study the duality theor…
The most important examples of a double vector bundle are provided by iterated tangent and cotangent functors: TTM, TT^*M, T^*TM, and T^*T^*M. We introduce the notions of the dual double vector bundle and the dual double vector bundle morphism. Theorems on canonical isomorphisms are formulated and proved. Several examp…
We describe the structure of the Lie groups endowed with a left-invariant symplectic form, called symplectic Lie groups, in terms of semi-direct products of Lie groups, symplectic reduction and principal bundles with affine fiber. This description is particularly nice if the group is Hamiltonian, that is, if the left c…
Even-dimensional simply connected manifolds that are rational homology spheres and double disk bundles are homeomorphic to spheres.
problem Characterizing manifolds that are both rational homology spheres and double disk bundles.
method Analyzing the structure of manifolds as unions of disk bundles and using properties of rational homology and cohomology.
result Even-dimensional simply connected manifolds that are rational homology spheres and double disk bundles are homeomorphic to spheres.
This paper studies covariant derivatives for Lie groupoids with representation-valued forms.
problem Understanding covariant derivatives for Lie groupoids with representation-valued forms.
method The paper explores two approaches: linear connections and multiplicative Ehresmann connections, both yielding geometrically richer curved double complexes.
result The horizontal exterior covariant derivative D is a key finding, generalizing the well-known operator from principal bundles. This paper equates geometric structures to algebraic ones for degree 2 manifolds.
problem Equivalence of geometric structures to algebraic ones for degree 2 manifolds.
method Using involutive double vector bundles and linear metrics, the paper establishes an equivalence between canonical categories.
result Equivalence of Poisson involutive double vector bundles to Poisson N-manifolds of degree 2.
The equivalence of principal bundles with transitive Lie groupoids due to Ehresmann is a well known result. A remarkable generalisation of this equivalence, due to Mackenzie, is the equivalence of principal bundle extensions with those transitive Lie groupoids over the total space of a principal bundle, which also admi…
Generalizes graded bundles with more flexible transformation laws.
problem No specific problem stated; focuses on generalization of graded bundles.
method Introduces filtered bundles with more general polynomial transformation laws.
result Linearisation of filtered bundles is well-defined.
Study Lie algebroid connections on principal bundles over complex projective varieties.
problem Existence and properties of Lie algebroid connections on principal bundles.
method Definition and study of Lie algebroid valued connections on holomorphic principal G-bundles, investigation of existence criteria.
result Investigation of criteria for existence of Lie algebroid connections on principal G-bundles over smooth complex projective curves.
Develops a unified theory of Yang-Mills and GR using generalized principal bundles.
problem Combining Yang-Mills theories and General Relativity into a single framework.
method Using generalized principal bundle theory, the authors develop a new approach to field theories.
result Recover General Relativity within the framework of generalized principal connections.
Holonomies match for higher local systems and principal 2-bundles.
problem Matching holonomies for higher local systems and principal 2-bundles.
method Higher Riemann-Hilbert correspondence and principal 2-bundles.
result Holonomies coincide for both formalisms.
Let X be a compact connected Kähler manifold equipped with an anti-holomorphic involution which is compatible with the Kähler structure. Let G be a connected complex reductive affine algebraic group equipped with a real form σG. We define pseudo-real principal G--bundles on X; these are generalizations of re…
This paper classifies equivariant principal bundles over a 2-sphere using isotropy representations.
problem Classifying equivariant principal bundles over the 2-sphere.
method Using isotropy representations to classify bundles over the 2-sphere.
result Equivariant principal bundles over the 2-sphere can be classified by a Γ-fixed set of homotopy classes of maps and first Chern class.
In this paper, we initiate the study of a parametrised version of Rieffel's strict deformation quantization. We apply it to give a classification of noncommutative principal torus bundles, in terms of parametrised strict deformation quantization of ordinary principal torus bundles. The paper also contains a putative de…
Defines hybrid systems on principal bundles and studies impact effects.
problem Understanding impact effects in hybrid mechanical systems.
method Defines hybrid systems on principal bundles, studies underlying geometry, and finds conditions for impact preservation.
result Conditions for preservation of both exterior and interior impacts by mechanical connections.
We define the pull-back of a smooth principal fibre bundle, and show that it has a natural principal fibre bundle structure. Next, we analyse the relationship between pull-backs by homotopy equivalent maps. The main result of this article is to show that for a principal fibre bundle over a paracompact manifold, there i…
Investigates connections adapted to a holomorphic Lie group action on bundles.
problem Finding connections adapted to a Lie group action on bundles.
method Analyzes connections on principal H-bundles over complex manifolds with holomorphic actions of Lie groups. result Identifies conditions for connections to be adapted to a given G-connection. A principal Higgs bundle (P,φ) over a singular curve X is a pair consisting of a principal bundle P and a morphism φ:X→AdP⊗ΩX1. We construct the moduli space of principal Higgs G-bundles over an irreducible singular curve X using the theory of decorated vector bundles. More precisely, given…
The study finds infinitely many counterexamples to a generalized Double Soul Conjecture.
problem The Double Soul Conjecture for non-simply connected manifolds.
method Analysis of double disk bundles and counterexamples.
result Infinitely many counterexamples to the generalized Double Soul Conjecture.
The paper explores grids and warps in double vector bundles, clarifying dual relationships.
problem Understanding dual relationships in double vector bundles and their iterated duals.
method Introduces squarecap sections and uses iterated duals to express warp of grids.
result An expression for the warp of a grid in terms of a pairing, extending dual relationships.
Gauge theory connects principal bundles and functors.
problem Understanding the relationship between principal bundles and functors.
method Characterized association functors and established an equivalence between principal bundles and functors.
result Established an equivalence between principal bundles and functors using association functors.
In this paper we introduce a notion of parallel transport for principal bundles with connections over differentiable stacks. We show that principal bundles with connections over stacks can be recovered from their parallel transport thereby extending the results of Barrett, Caetano and Picken, and Schreiber and Waldof f…
Paper introduces Atiyah sequence for Lie groupoids and studies gauge transformations.
problem Defining connections on principal 2-bundles over Lie groupoids.
method Introduced Atiyah sequence, defined strict and semi-strict connections, constructed gauge transformations.
result Existence criterion for connections on principal 2-bundles over proper, étale Lie groupoids.
Hyperbolic surface bundles over the circle have minimal entropy close to 1/g.
problem Understanding the minimal entropy of hyperbolic surface bundles over the circle.
method Proved the minimal entropy behavior of hyperbolic surface bundles as branched double covers of the 3-sphere.
result Minimal entropy of hyperbolic surface bundles over the circle behaves like 1/g.
The paper studies spectral convergence of connections on vector and principal bundles.
problem Continuity of eigenvalues of connection Laplacians on vector bundles.
method Introducing a new topology on metric measure spaces with isometric G-actions to analyze convergence of G-connections. result Established spectral convergence of connections on vector and principal bundles.
Study characterizes martingales on fiber bundles for harmonic map analysis.
problem Characterizing martingales on fiber bundles for harmonic maps.
method Investigated ablaP-martingales on principal fiber bundles P(M,G) with a projectable connection. result New characterizations of harmonic maps established.
Researchers describe how special conic bundles deform into double solids.
problem Understanding the versal deformation of conic bundles over 3CP2. method Explicit description of deformation in a general context.
result Explicit description of the deformation of conic bundles into double solids.
Constructs parallel transport in higher gauge theory using principal 2-bundles.
problem Higher gauge theory and parallel transport in categorified principal bundles.
method Explicit construction of parallel transport for connections on principal 2-bundles.
result Proves constructions fit into axiomatic framework for categorified parallel transport.
Logarithmic connections on principal bundles over normal varieties are studied.
problem Existence and properties of logarithmic connections on principal bundles over normal varieties.
method Introducing logarithmic connections, showing equivalence to covariant derivatives, and proving existence conditions.
result Existence of logarithmic connections on principal bundles over normal varieties is equivalent to certain conditions on the associated vector bundles and adjoint bundles.
Smooth classifying spaces for groups defined using diffeological spaces.
problem Classifying smooth principal bundles for a smooth group G. method Developed the theory of smooth principal bundles using diffeological spaces, defining D-numerable bundles and proving classification results. result Smooth structures on Milnor's spaces EG and BG classify all D-numerable principal bundles over any diffeological space. The paper classifies Lie algebroids and their connections, modulating principal objects.
problem Classifying and studying Lie algebroids and their connections.
method Classifying integrable transitive Lie algebroids, introducing Higgs bundles, and using Tannakian categories.
result Moduli spaces of principal \(G\)-bundles and Higgs bundles are semiprojective varieties.