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

Grabowski and Rotkiewicz showed that a smooth action of the monoid of multiplicative reals equips a manifold with the structure of a, so called, graded bundle, being a natural generalization of the notion of a vector bundle. In the light of this result it is interesting to ask what are the natural structures on manifolds related with actions of other natural monoids generalizing the multiplicative reals. We shall discuss this problem in a series of two lectures. In the first of these (M. Jóźwikowski, 23. March) we will recall the known results of Grabowski and Rotkiewicz and discuss actions of the monoid of second jets of real functions which preserve zero. In the second (M.Rotkiewicz, 30. March) we will discuss actions of the multiplicative complex numbers and actions of the multiplicative reals on supermanifolds.