Graded bundles and homogeneity structures

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 local coordinates are homogeneous functions of degrees 0,1,...,n. We prove that graded bundles have a convenient equivalent description as homogeneity structures, i.e. manifolds with a smooth action of the multiplicative monoid of non-negative reals. The main result states that each homogeneity structure admits an atlas whose local coordinates are homogeneous. Considering a natural compatibility condition of homogeneity structures we formulate, in turn, the concept of a double (r-tuple, in general) graded bundle - a broad generalization of the concept of a double (r-tuple) vector bundle. Double graded bundles are proven to be locally trivial in the sense that we can find local coordinates which are simultaneously homogeneous with respect to both homogeneity structures.Comment: 19 pages, the revised version to be published in J. Geom. Phy

Higher vector bundles and multi-graded symplectic manifolds

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 literature can be equivalently defined as manifolds with a family of commuting Euler vector fields. Higher vector bundles can be therefore defined as manifolds admitting certain $\N^n$-grading in the structure sheaf. Consequently, multi-graded (super)manifolds are canonically associated with higher vector bundles that is an equivalence of categories. Of particular interest are symplectic multi-graded manifolds which are proven to be associated with cotangent bundles. Duality for higher vector bundles is then explained by means of the cotangent bundles as they contain the collection of all possible duals. This gives, moreover, higher generalizations of the known `universal Legendre transformation' T*E->T*E*, identifying the cotangent bundles of all higher vector bundles in duality. The symplectic multi-graded manifolds, equipped with certain homological Hamiltonian vector fields, lead to an alternative to Roytenberg's picture generalization of Lie bialgebroids, Courant brackets, Drinfeld doubles and can be viewed as geometrical base for higher BRST and Batalin-Vilkovisky formalisms. This is also a natural framework for studying n-fold Lie algebroids and related structures.Comment: 27 pages, minor corrections, to appear in J. Geom. Phy