Related papers: A topological definition of the Maslov bundle
We develop a coarse notion of bundle and use it to understand the coarse geometry of group extensions and, more generally, groups acting on proper metric spaces. The results are particularly sharp for groups acting on (locally finite) trees…
This paper studies the homotopy theory of parameterized spectrum objects in a model category from a global point of view. More precisely, for a model category $\mathcal{M}$ satisfying suitable conditions, we construct a relative model…
We introduce a homology theory whose Euler characteristics counts ASD bundles over four dimensional co-associative submanifolds in (almost) G_2 manifolds. As a TQFT, in relative situations, we have the Fukaya-Floer category of Lagrangians…
We prove that the higher direct images of the dualizing sheaf of a Lagrangian fibration between smooth projective manifolds are isomorphic to the cotangent bundles of base space. As a corollary, we obtain that every Hodge number of the base…
We present the notion of higher Kirillov brackets on the sections of an even line bundle over a supermanifold. When the line bundle is trivial we shall speak of higher Jacobi brackets. These brackets are understood furnishing the module of…
We present some plausible definitions for the tangent grupoid of a manifold M, as well as some of the known applications of the structure. This is a kind of introductory note.
It is proved that the Maslov index naturally arises in the framework of PDEs geometry. The characterization of PDE solutions by means of Maslov index is given. With this respect, Maslov index for Lagrangian submanifolds is given on the…
The sub-Finslerian geometry means that the metric $F$ is defined only on a given subbundle of the tangent bundle, called a horizontal bundle. In the paper, a version of the Hopf-Rinow theorem is proved in the case of sub-Finslerian…
For an arbitrary submanifold $M \subset \mathbb{C}P^n$ we determine conditions under which it is austere, i.e., the normal bundle of $M$ is special Lagrangian with respect to Stenzel's Ricci-flat K\"ahler metric on $T\mathbb{C}P^n$. We also…
Tangent categories are categories equipped with a tangent functor: an endofunctor with certain natural transformations which make it behave like the tangent bundle functor on the category of smooth manifolds. They provide an abstract…
Higher bundles are homotopy coherent generalisations of classical fibre bundles. They appear in numerous contexts in geometry, topology and physics. In particular, higher principal bundles provide the geometric framework for higher-group…
We establish a new version of Floer homology for monotone Lagrangian submanifolds and apply it to prove the following (generalized) version of Audin's conjecture : if $L$ is an aspherical manifold which admits a monotone Lagrangian…
Given a vector bundle $A\to M$ we study the geometry of the graded manifolds $T^*[k]A[1]$, including their canonical symplectic structures, compatible Q-structures and Lagrangian Q-submanifolds. We relate these graded objects to classical…
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.…
The notion of a higher bundle gerbe is introduced to give a geometric realization of the higher degree integral cohomology of certain manifolds. We consider examples using the infinite dimensional spaces arising in gauge theories.
We generalize Kontsevich's construction of characteristic classes of fibre bundles with homology sphere fibres and a trivialization of the vertical tangent bundle to framed fibre bundles with closed manifold fibres.
A linear F-manifold is an F-manifold (E, \circ , e) defined on the total space of a vector bundle \pi : E \rightarrow M for which the multiplication and unit field are linear tensor fields. We develop a systematic treatment of linear…
The coupling of the tangent bundle $TM$ with the Lie algebra bundle $L$ (K.Mackenzie,2005, Definition 7.2.2) plays the crucial role in the classification of the transitive Lie algebroids for Lie algebra bundle $L$ with fixed finite…
We describe the notion of a \emph{weighting} along a submanifold $N\subset M$, and explore its differential-geometric implications. This includes a detailed discussion of weighted normal bundles, weighted deformation spaces, and weighted…
Inspired by the recent algebraic approach to classical field theory, we propose a more general setting based on the manifold of smooth sections of a non-trivial fiber bundle. Central is the notion of observables over such sections, i.e.…