Affine connections, and midpoint formation

Abstract

It is a striking fact that differential calculus exists not only in analysis (based on the real numbers ), but also in algebraic geometry, where no limit processes are available. In algebraic geometry, one rather uses the idea of nilpotent elements in the "affine line" R; they act as infinitesimals. (Recall that an element x in a ring R is called nilpotent if x k ∈=∈0 for suitable non-negative integer k.) Synthetic differential geometry (SDG) is an axiomatic theory, based on such nilpotent infinitesimals. It can be proved, via topos theory, that the axiomatics covers both the differential-geometric notions of algebraic geometry and those of calculus. © 2009 Springer Berlin Heidelberg.