Seminarium "The Trans-Carpathian Seminar on Geometry & Physics"

Abstract: Graded bundles can be viewed as a natural generalization of vector bundles. In short, they are locally trivial fibered bundles with fibers possessing a structure of a graded space, i.e. a manifold diffeomorphic to Rⁿ with a distinguished class of global coordinates with positive integer weights assigned. In a special case when these weights are all equal to 1, a graded space becomes a standard vector space and a graded bundle - a vector bundle. The fundamental example of a graded bundle isthe k-th order tangent bundle of a manifold M. Can we turn graded bundles in the realm of vector bundles? We shall construct a functor which takes a graded bundle of degree k and produces a k-fold vector bundle, mimicking the the canonical embedding T^kM into TT...T M. Can we linearise other graded structures in similar way? What is the image of the linearisation functor? Similar question will be discussed. Based on a joint paper with A. J. Bruce and J. Grabowski