A note on actions of some monoids

Abstract

Smooth actions of the multiplicative monoid $(\mathbb{R},\cdot )$ of real numbers on manifolds lead to an alternative, and for some reasons simpler, definition of a vector bundle, a double vector bundle and related structures like a graded bundle [Grabowski and Rotkiewicz, J. Geom. Phys. 2011]. For these reasons it is natural to study smooth actions of certain monoids closely related with the monoid $(\mathbb{R},\cdot )$ . 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 $(\mathbb{R},0)\to (\mathbb{R},0)$, smooth action of the monoid of real 2 by 2 matrices and smooth actions of 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.

Final version published in DGA. Corrected proof of Theorem 3.18. Watch our publication: http://audioslides.elsevier.com//ViewerLarge.aspx?source=1&doi=10.1016/j.difgeo.2016.04.003

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