在数学中有个课题是微分几何，其中仿射联系是个定义在流形上的几何对象，连接了邻近几点上的切空间，使得在流形上的切矢量场可以求导。仿射联系的概念起源于19世纪的几何学和张量微积分，但那时并没有被完备的定义出来。直到1920年，（用于Cartan connection理论）及Hermann Weyl（做为广义相对论的基础理论）。这专门术语是沿用Cartan所使用的术语及根据从欧几里德空间R中切空间的推广。换句话说，仿射联系的概念是为了推广欧几里德空间，使得流形上每点都有一个光滑的（可无限求导）仿射空间。

In the branch of mathematics called differential geometry, an affine connection is a geometric object on a smooth manifold which connects nearby tangent spaces, and so permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space. The notion of an affine connection has its roots in 19th-century geometry and tensor calculus, but was not fully developed until the early 1920s, by Élie Cartan (as part of his general theory of connections) and Hermann Weyl (who used the notion as a part of his foundations for general relativity). The terminology is due to Cartan and has its origins in the identification of tangent spaces in Euclidean space R by translation: the idea is that a choice of affine connection makes a manifold look infinitesimally like Euclidean space not just smoothly, but as an affine space.