# The origin of the name “connection” in differential geometry

Everyone will encounter the notion of connection in differential geometry. But who gave this name of connection( or affine connection)? Why is this derivative operator called connection? What object does it connect? I'm just confused about the origin of the name. Thanks!

• What object does it connect? It connects nearby (local) tangent spaces on a smooth manifold. It is used to show the changes to tangents as a result of parallel transport. It is usually coordinate based. I have no idea though who coined the term but it seems to be a good fitting obvious name. – K7PEH Jun 7 '15 at 15:54

According to Freeman's Historical Overview of Connections in Geometry Hermann Weyl introduced the term "connection" in his 1918 text Reine Infinitesimal Geometrie. His definition explains the connection part, affine connection is "that which determines into which vector at $P'$ a vector at an infinitesimally close point $P$ will transform under parallel displacement from $P$ to $P'$". The affine part comes from Weyl requiring that the "parallel displacement" be an affine transformation, i.e. preserve collinearity and ratios of distances but not necessarily distances themselves, from "totality of vectors" at $P$ to that at $P'$. The year before Levi-Civita (and independently Hessenberg and Schouten) introduced the notion of parallel directions and gave a geometric interpretation of the Riemann curvature in terms of parallel transport, writing "once the law is known according to which one passes from a point to a point infinitely close to it, one is able immediately to accomplish the displacement of parallel directions along any arbitrary curve $C$". This is the first expression of the idea, and connections associated to Riemannian metrics are now called Levi-Civita connections.