Constructs a functor for graded bundles to vector bundles, characterizing symmetric structures.
problem Characterizing and fully characterizing the image of a functor from graded bundles to vector bundles.
method Constructs a full linearisation functor that takes a graded bundle of degree k to a k-fold vector bundle, fully characterizing the image and discussing related cases.
result Obtains a subcategory of k-fold vector bundles consisting of symmetric k-fold vector bundles equipped with a family of morphisms indexed by the symmetric group Sk. A natural explicit condition is given ensuring that an action of the multiplicative monoid of non-negative reals on a manifold F comes from homotheties of a vector bundle structure on F, or, equivalently, from an Euler vector field. This is used in showing that double (or higher) vector bundles present in the literatur…
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.
Study on involution in double jet bundles.
problem Exploring involution in double jet bundles.
method Generalized double tangent bundles to double jet bundles, presented secondary vector bundle structure, proved involution.
result Existence of a natural involution on double jet bundles.
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.
These notes are an introduction to symplectic groupoids and the double structures associated with them. The treatment is intended to lie about midway between the original account of Coste, Dazord and Weinstein, which relied on effective use of the symplectic structures, and the account in my 2005 book, which showed, on…
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…
The Lagrangian description of mechanical systems and the Legendre Transformation (considered as a passage from the Lagrangian to the Hamiltonian formulation of the dynamics) for point-like objects, for which the infinitesimal configuration space is TM, is based on the existence of canonical symplectic isomorphisms of d…
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.
Explores Lie theory in vector bundles and Poisson geometry.
problem Understanding Lie theory in vector bundles and Poisson geometry.
method Review of Lie theory in vector bundles over Lie groupoids and algebroids, extending to double Lie algebroids and LA-groupoids.
result Extension of Lie theory to double Lie algebroids and LA-groupoids using Poisson geometry.
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.
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…
Every compact symplectic 4-manifold can be realized as a branched cover of the complex projective plane branched along a symplectic curve with cusp and node singularities; the covering map is induced by a triple of sections of a "very ample" line bundle. In this paper, we give an explicit formula describing the behavio…
On a Poisson manifold endowed with a Riemannian metric we will construct a vector field that generalizes the double bracket vector field defined on semi-simple Lie algebras. On a regular symplectic leaf we will construct a generalization of the normal metric such that the above vector field restricted to the symplectic…
The canonical involution of a double (=iterated) tangent bundle may be dualized in different ways to yield relations between the Tulczyjew diffeomorphism, the Poisson anchor associated with the standard symplectic structure on the cotangent space,and the reversal diffeomorphism. We show that the constructions which yie…
Double Lie algebroids were discovered by Kirill Mackenzie from the study of double Lie groupoids and were defined in terms of rather complicated conditions making use of duality theory for Lie algebroids and double vector bundles. In this paper we establish a simple alternative characterization of double Lie algebroids…
Quantizes symplectic fibrations to analyze vector bundles and metrics.
problem Quantizing higher rank vector bundles and understanding their metrics.
method Relates Berezin-Toeplitz quantization to hybrid systems and symplectic fibrations.
result Established refined estimates for computing balanced metrics on Kähler manifolds.
We obtain an explicit formula for the symplectic form over the double quotient with help of the Green function of a Riemann surface.
This paper establishes a correspondence between Einstein-Hermitian metrics and stability for generalized holomorphic vector bundles over symplectic type generalized Kahler manifolds.
problem Establishing a relationship between Einstein-Hermitian metrics and stability for generalized holomorphic vector bundles.
method Using the moment map framework and Poisson modules, the authors prove the Kobayashi-Hitchin correspondence.
result The equivalence of the existence of an Einstein-Hermitian metric and ψ-polystability of a generalized holomorphic vector bundle. Investigates metric degeneracies on symplectic leaves using a generalized gradient flow.
problem Degeneracies in metrics on symplectic leaves of Poisson manifolds.
method Introduces the generalized double bracket (GDB) vector field to generalize gradient dynamics.
result Identifies admissible regions where the double bracket metric remains non-degenerate on symplectic leaves, enabling GDB as a gradient flow.
This paper studies graded manifolds of type Δ and their equivalence with n-fold vector bundles.
problem Understanding the relationship between graded manifolds and vector bundles.
method Geometrization process for Zr-graded manifolds of type Δ. result Established an equivalence between a subcategory of n-fold vector bundles and graded manifolds of type Δ.
The word `double' was used by Ehresmann to mean `an object X in the category of all X'. Double categories, double groupoids and double vector bundles are instances, but the notion of Lie algebroid cannot readily be doubled in the Ehresmann sense, since a Lie algebroid bracket cannot be defined diagrammatically. In this…
Enhances Hamiltonian systems stability through generalized double bracket vector fields.
problem Stabilizing already stable points in Hamiltonian systems.
method Generalized double bracket vector fields on Poisson manifolds with pseudo-Riemannian metrics.
result Enhanced equilibria stability through dissipation terms.
We define a new algebra for double vector bundles, linking it to Lie algebroids.
problem Understanding structures on double vector bundles and Lie algebroids.
method Defining a Weil algebra for double vector bundles and relating it to Lie algebroids.
result Double Lie algebroid structures are characterized by Gerstenhaber brackets on the Weil algebra.
Extends Drinfel'd doubles to include twists and generalizes Lie bialgebroids.
problem Tackles the extension of Drinfel'd doubles and Lie bialgebroids to include twists and generalizes them.
method Introduces a framework of calculus on algebroids and examines compatibility conditions for various algebroid properties.
result Introduces the notion of proto bialgebroids and their Drinfel'd doubles, generalizing both bialgebroids and proto Lie bialgebroids.
We study noncommutative generalizations of such notions of the classical symplectic geometry as degenerate Poisson structure, Poisson submanifold and quotient manifold, symplectic foliation and symplectic leaf for associative Poisson algebras. We consider these structures for the case of the endomorphism algebra of a v…
Summarizes geometric formulation of Einstein-Scalar-Maxwell theories.
problem Geometric formulation of Einstein-Scalar-Maxwell theories.
method Global geometric formulation using flat symplectic vector bundles.
result Describes scalar-electromagnetic symmetry group and Dirac quantization condition.
Symplectic 4-manifolds with Kodaira dimension zero can be viewed as symplectic Calabi-Yau surfaces. We are able to completely determine their Betti numbers by proving two general results on quaternionic vector bundles.
Geometric structures on NQ-manifolds, i.e.~non-negatively graded manifolds with an homological vector field, encode non-graded geometric data on Lie algebroids and their higher analogues. A particularly relevant class of structures consists of vector bundle valued differential forms. Symplectic forms, contac…
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.
We define \textit{graded manifolds} as a version of supermanifolds endowed with an additional Z-grading in the structure sheaf, called \textit{weight} (not linked with parity). Examples are ordinary supermanifolds, vector bundles over supermanifolds, double vector bundles, iterated constructions like TTM, e…
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…
New differential complexes on symplectic manifolds.
problem Developing calculus on symplectic manifolds.
method Coupling a symplectic manifold to a vector bundle with a constrained curvature.
result Construction of new differential complexes.
We introduce the concept of a graded bundle which is a natural generalization of the concept of a vector bundle and whose standard examples are higher tangent bundles T^nQ playing a fundamental role in higher order Lagrangian formalisms. Graded bundles are graded manifolds in the sense that we can choose an atlas whose…
The paper explores dual structures in graded manifolds.
problem Understanding dual objects in graded manifolds.
method Developed dual objects for graded bundles and applied them to double vector bundles and graded bundles of degree 2.
result Elegant characterizations of double vector bundles and graded bundles of degree 2.
The paper geometrizes N-manifolds using symmetric vector bundles.
problem Geometrizing N-manifolds with a specific symmetry.
method Equivalence between [n]-manifolds and symmetric n-fold vector bundles. result Identical cocycles between symmetric vector bundles and [n]-manifolds. Study of generalized double Bruhat cells and their integrations.
problem Understanding and integrating generalized double Bruhat cells in Lie groups.
method Integrating Poisson groupoids to symplectic double groupoids, relating to fission spaces of irregular singularities.
result Explicit integrations of Poisson groupoids and Morita equivalence of double groupoids.
We propose a construction of Kähler and non-Kähler Calabi-Yau manifolds by branched double covers of twistor spaces. In this construction we use the twistor spaces of four-manifolds with self-dual conformal structures, with the examples of connected sum of n P2s. We also construct K3-fibered Calabi-Ya…
In this paper we obtain the following results: (1) Any compact Stein surface with boundary embeds naturally into a symplectic Lefschetz fibration over the 2-sphere. (2) There exists a minimal elliptic fibration over the 2-disk, which is not Stein. (3) The circle bundle over a genus n>1 surface with euler number e=-1 ad…
This paper presents a generalization of symplectic geometry to a principal bundle over the configuration space of a classical field. This bundle, the vertically adapted linear frame bundle, is obtained by breaking the symmetry of the full linear frame bundle of the field configuration space, and it inherits a generaliz…
On a complex manifold, a co-Higgs bundle is a holomorphic vector bundle with an endomorphism twisted by the tangent bundle. The notion of generalized holomorphic bundle in Hitchin's generalized geometry coincides with that of co-Higgs bundle when the generalized complex manifold is ordinary complex. Schwarzenberger's r…
The article investigates conditions for isomorphism of singular tangent bundles.
problem Conditions for isomorphism of singular tangent bundles.
method Logarithmic and b-tangent bundles approach to resolve singularities. result Established a Poincaré-Hopf theorem for bm-tangent bundles. Unified Jacobi coupling construction for various geometric settings.
problem Constructing Jacobi structures on associated bundles.
method Extending Sternberg--Weinstein coupling to Jacobi geometry.
result Associated bundles inherit Jacobi structures from base spaces.
A geometric quantization of a Kähler manifold, viewed as a symplectic manifold, depends on the complex structure compatible with the symplectic form. The quantizations form a vector bundle over the space of such complex structures. Having a canonical quantization would amount to finding a natural (projectively) flat co…
The abstract discusses the equivalence of transnormal and isoparametric functions on compact manifolds.
problem The existence of transnormal and isoparametric functions on compact manifolds.
method Exploring embedded transnormal systems and showing the existence of transnormal functions on Riemannian manifolds.
result Compact manifolds with transnormal functions also have isoparametric functions, and vice versa.
Introduces symplectic hopfoids and their relation to double Lie groupoids.
problem Understanding symplectic structures in groupoid-like objects.
method Introduces symplectic hopfoids and shows their correspondence to symplectic double groupoids.
result Symplectic hopfoids and double Lie groupoids are equivalent.
Symplectic structure found on moduli space of framed Higgs bundles.
problem Understanding the symplectic structure of moduli spaces of Higgs bundles.
method Trivializing vector bundles over divisors to construct framed Higgs bundles, proving symplectic structure.
result Moduli space of framed Higgs bundles admits a natural holomorphic symplectic structure.