Vector fields on R^R in well adapted models of synthetic differential geometry

Abstract

Vector fields in infinite-dimensional manifolds play an important role in differential topology-geometry. In particular the case when the manifolds are C∞-map spaces. The well developed theory modeled in Banach spaces does not apply here. Instead a theory modeled in Frechet spaces is being considered. This is a theory which seems to be a much less straightforward generalization of the finite-dimensional case. The well adapted models of S.D.G. lead naturally to treat these spaces. We investigate here the case of the 'manifold' R^R, whose space of global sections is C^∞(ℝ). We prove that to integrate a vector field in R^R is equivalent to a certain differential problem in C^∞(ℝ). To do this, we previously characterize the maps R^R → R^R, R^R X R^R in the topos by means of the functions they induce in the respective spaces of global sections.