Differential Geometry over General Base Fields and Rings

The aim of this work is to lay the foundations of differential geometry and Lie theory over the general class of topological base fields and -rings for which a differential calculus has been developed in [BGN04], without any restriction on the dimension or on the characteristic. Two basic features distinguish our approach from the classical real (finite or infinite dimensional) theory, namely the interpretation of tangentand jet functors as functors of scalar extensions and the introduction of multilinear bundles and multilinear connections which generalize the concept of vector bundles and linear connections.