Sections along Maps in Geometry and Physics

Talk given at the meeting Geometry and Physics on the occasion of the 65th birthday of W.M. Tulczyjew 1 1. Introduction. Geometric techniques have been applied to physics in many different ways and they have provided powerful methods of dealing with classical problems from a new geometric perspective. Vector fields, forms, exterior differential calculus, Lie groups, fibre bundles, connections, etc..., are now well established tools in modern physics. One of the main contributions of Prof. W.M. Tulczyjew is the alternative way of using the geometric concepts in a more algebraic approach, using, for instance, derivations of algebras and related concepts. The aim of this paper is to point out some of my contributions in this direction [1–3,6], which have received a big influence of Tulczyjew's works [7,8,9]. Moreover, it will be shown how this approach allows us to translate the usual concepts arising in Geometrical Mechanics to the framework of Supermechanics.