The affine bundle theorem in synthetic differential geometry of jet bundles

Abstract

In our [Higher-order preconnections in synthetic differential geometry of jet bundles, Beitr\"{a}ge zur Algebra und Geometrie, 45 (2004), 677-696] we have established the affine bundle theorem in the synthetic approach to jet bundles in terms of infinitesimal spaces Dⁿ's. In our succeeding [Synthetic differential geometry of higher-order total differentials, to appear in Cahiers de Topologie et G\'{e}om\'{e}trie Diff\'{e}rentielle Cat\'{e}goriques] we have introduced another synthetic approach to jet bundles in terms of infinitesimal spaces D_{n}'s, and have compared it with the former approach both in the general microlinear setting and in the finite-dimensional setting. However our comparison in the finite-dimensional setting was incomplete, for our use of dimension counting techniques has tacitly assumed that jet bundles in terms of Dⁿ's and D_{n}'s are vector bundles, which is not the case unless the given bundle is already a vector bundle. The principal objective in this paper is to establish the affine bundle theorem in our second synthetic approach to jet bundles and then to compare not merely the jet bundles based on both approaches but the affine bundles constructed as a whole both in the general microlinear setting and in the finite-dimensional setting. This completes our comparsion between the two approaches in the finite-dimensional setting.