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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!

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    $\begingroup$ 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. $\endgroup$ – K7PEH Jun 7 '15 at 15:54
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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.

Formally Riemannian connections appeared much earlier, but relation to parallelism remained unknown. Back in 1869 Christoffel introduced the symbols now named after him, that in modern terminology represent coefficients of a connection. However, he was interested only in classifying invariants and "conditions that differential expressions of degree 2 must verify to be able to transform one into another by a change of variables", not in any geometric interpretations. In 1890s Ricci-Curbastro noticed that because of their transformation properties Christoffel symbols of a Riemannian metric could be used to create an "absolute differential calculus", now known as tensor calculus, but he used the terms "covariant and contravariant derivatives", not "connections". In 1901 Ricci and his student Levi-Civita wrote a long expose of the new calculus, but according to Levi-Civita, "for many more years it was used almost exclusively by its inventor and a few of his students", until in general relativity "Ricci's calculus revealed itself to be not only useful but truly indispensable". By the way, Levi-Civita corresponded with Einstein extensively after 1916 helping him with the technical side of tensor calculus as it applies to general relativity.

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    $\begingroup$ When you say "[...] gave a geometric interpretation of the Riemann curvature," do you mean that Riemann didn't have a geometric motivation for defining the Riemann curvature tensor? $\endgroup$ – hjhjhj57 Jun 15 '15 at 7:31
  • $\begingroup$ @Javier Good point, I tightened up the language. Riemann does not give an interpretation in terms of change of direction under transport around a loop. He talks about "surface-direction" (what we would call a plane in tangent space) and prolonging all geodesics originally tangent to it, which creates a surface. This surface has total curvature, and he refers to Gauss's interpretation of it in terms of spherical excess, or area excess for infinitesimal triangles. cs.jhu.edu/~misha/ReadingSeminar/Papers/Riemann54.pdf $\endgroup$ – Conifold Jun 15 '15 at 22:25
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If you look at Weyl's 1918 Raum, Zeit, Materie, you will see in the index three kinds of connection listed under the word "Zusammenhang", which we now translate as "connection". His three kinds of connections were continuous, affine and metric. Weyl clearly thought of the affine connection as an intermediate stage of connectivity between topological connectivity and the full Riemannian metric.

Ricci and Levi-Civita had introduced the covariant derivative much earlier, about 1900/1901, but Levi-Civita did not talk about connections even in his 1925/1926 book on the absolute differential calculus. Levi-Civita's first mention of parallel transport was in a 1917 paper "Nozione di parallelismo ...", but he did not use any word like "connection". Since we nowadays identify parallel transport with an affine (or other kind of) connection, it could be said that the notion of a connection originated in 1917 with Levi-Civita. Weyl directly acknowledge Levi-Civita for this on page 94 of "Raum, Zeit, Materie". But it was Weyl who coined the term "Connection" ("Zusammenhang") by analogy with topological connectivity to mean the idea of local parallel transport of vectors. Thus the motivation for the name "connection" was the analogy with topological connectivity.

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