A contribution to the connection theory of areal spaces

In this lecture we want to discuss connections in areal spaces. An areal space is a manifold endowed with a general notion of area. In order to make this notion natural and to show its origin we start with Finsler spaces which are built on a general notion of the arclength. A connection in an areal space is a homogeneous connection in a fibre bundle. We show that this can be replaced by a nonlinear connection in a much simpler vector bundle. Finally, we apply our previous considerations in order to give an answer to the question of the Riemannian metrizability of an affine connection.