A note on actions of some monoids

Smooth actions of the multiplicative monoid (R,⋅) of real numbers on manifolds lead to an alternative, and for some reasons simpler, definitions of a vector bundle, a double vector bundle and related structures like a graded bundle (Grabowski and Rotkiewicz (2011) [10]). For these reasons it is natural to study smooth actions of certain monoids closely related with the monoid (R,⋅). Namely, we discuss geometric structures naturally related with: smooth and holomorphic actions of the monoid of multiplicative complex numbers, smooth actions of the monoid of second jets of punctured maps (R,0)→(R,0), smooth actions of the monoid of real 2 by 2 matrices and smooth actions of the multiplicative reals on a supermanifold. In particular cases we recover the notions of a holomorphic vector bundle, a complex vector bundle and a non-negatively graded manifold.