Affine connections and second-order affine structures

TL;DR

This paper explores how smooth manifolds can be endowed with higher-order infinitesimal affine structures, extending the concept of affine connections to second-order neighborhoods within the framework of Synthetic Differential Geometry.

Contribution

It introduces natural infinitesimal structures for higher-order neighborhoods and demonstrates that any symmetric affine connection extends to a second-order infinitesimally affine structure.

Findings

Constructed natural infinitesimal structures for higher-order neighborhoods

Extended symmetric affine connections to second-order structures

Unified first and second-order infinitesimal affine geometries

Abstract

Smooth manifolds have been always understood intuitively as spaces with an affine geometry on the infinitesimal scale. In Synthetic Differential Geometry this can be made precise by showing that a smooth manifold carries a natural structure of an infinitesimally affine space. This structure is comprised of two pieces of data: a sequence of symmetric and reflexive relations defining the tuples of mutual infinitesimally close points, called an infinitesimal structure, and an action of affine combinations on these tuples. For smooth manifolds the only natural infinitesimal structure that has been considered so far is the one generated by the first neighbourhood of the diagonal. In this paper we construct natural infinitesimal structures for higher-order neighbourhoods of the diagonal and show that on any manifold any symmetric affine connection extends to a second-order infinitesimally affine structure.