PART II - TOPICS IN SDG

Summary

In this second part we illustrate the principles of synthetic differential geometry and topology in two distinct areas. The first example is a theory of connections and sprays, where we show that—unlike the classical situation—the passage from connections to geodesic sprays need not involve integration, except in infinitesimal form. Moreover, the validity of the Ambrose-Palais-Singer theorem within SDG extends well beyond the classical one. In our second example we show how in SDG one can develop a calculus of variations ‘without variations’, except for those of an infinitesimal nature. Once again, the range of applications of the calculus of variations within SDG extends beyond the classical one. Indeed, in both examples, we work with infinitesimally linear objects—a class closed under finite limits, exponentiation, and ´etale descent. The existence of well adapted models of SDG guarantees that those theories developed in its context are indeed relevant to the corresponding classical theories.