Differential Geometry on Pointwise Affine Spaces

We introduce an alternative formalization of curved spaces in which the concept of a pointwise affine space, as defined here, replaces that of a manifold. New or modified definitions of familiar notions from differential geometry such as connections, torsion, Riemann curvature, vector fields and differentiation of vector fields are presented, and examples of applications of the resulting framework are given. In general, the notions and results considered receive simpler and more transparent forms than in conventional approaches.