26 votes
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How was Einstein led to make a contact with Differential Geometry for his theory of General Relativity?

Einstein himself told the story in his Kyoto address of 1922, which I quote from Pais's biography titled Subtle is the Lord: "If all systems are equivalent, then Euclidean geometry cannot hold in ...
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18 votes

What was the motivation for the development of modern, intrinsic, differential geometry?

Tensor calculus was developed about 20 years before the general relativity by Ricci and Levi-Civita, starting around 1890, under the name of absolute differential calculus. It was motivated by Riemann'...
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17 votes

When and how was the geometric understanding of gauge theories developed?

I will focus on the history before the Yang-Mills paper. The first harbinger was the introduction of the scalar potential for the gravitational field by Lagrange in 1773. In 1864 Maxwell introduced ...
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17 votes
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When and how was the geometric understanding of gauge theories developed?

I'll focus on the geometry of Yang-Mills theories specifically, but as Conifold's answer points out, gauge theories were studied geometrically long before the work of Yang and Mills. The foreward to ...
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15 votes

How come we attribute the general theory of relativity to Einstein?

This seems a bit of a naive question to me. Einstein had been working on this problem for several years (starting as early as 1907), and had developed much of the physics by 1912. He greatly struggled ...
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12 votes
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How was curvature originally defined and calculated?

Apollonius (c. 262–190 BC) "calculated" curvature of conic sections implicitly when solving the problem of drawing normals to them in book V of Conica, but he did not think of it as a ...
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11 votes
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Who do I blame for non-Euclidean geometry?

Technically, the first was Lobachevski (published in 1829-30). Bolyai was independent (published in 1832). Gauss discovered it independently of both (not published). A more complicated research is ...
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11 votes
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Help translate from German a quote by Hermann Weyl in Space Time Matter

One must firmly protest these orgies of a formalism with which even technicians are getting harassed today. (Literally: of the formalism, with which one is beginning to harass even technicians today.)...
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9 votes

How come we attribute the general theory of relativity to Einstein?

First, to echo and elaborate on previous answers, Hilbert only appeared on the scene during the last few minutes of the last act, so to speak. Einstein made his first step towards the General Theory ...
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9 votes

Was Euler's theorem in differential geometry motivated by matrices and eigenvalues?

You can read about the origin of Euler's 1760 study on curvature in Mathematical Masterpieces: Further Chronicles by the Explorers (page 187 and following). The idea of curvature goes back several ...
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9 votes
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Why are the symbols E, F, G, L, M, and N used for the coefficients of the fundamental forms?

See the paper by Gauss, Disquisitiones generales circa superficies curvas, from 1828. On page 24, we find: $$ E\,dp^2 + 2F\,dp\,dq + G\,dq^2 . $$ Since then everyone continued to use the letters $E,F,...
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8 votes
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What is the origin of the use of "g" for a Riemannian metric?

You are guessing correctly. Riemann did not use $g$ for the metric tensor, he writes things like $ds^2$ or $\sum dx^2$ instead, see his 1854 lecture "On the Hypotheses which lie at the Bases of ...
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8 votes
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How did the exponential map of Riemannian geometry get its name?

History does not often develop in the order of textbook expositions. Today the exponential map is introduced early in both Riemannian geometry and Lie group theory, but many results it is used to ...
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8 votes
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Why is distance sometimes abbreviated S?

I think it most likely stands for spatium. E.g. Euler’s first book Mechanica (1736) uses $s$ throughout and first introduces it as follows (p. 13): Theorema. (...) oportet determinare tempus, quo ...
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8 votes

What is the history of staircase or 𝜋=4 paradox?

The "staircase paradox" (or "Pythagoras paradox") name appears to be recent, so it is hard to search for it. Wolfram calls it "diagonal paradox", but that may be conflating it with a different paradox ...
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7 votes
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The origin of the name "connection" in differential geometry

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 ...
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7 votes
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Was Euler's theorem in differential geometry motivated by matrices and eigenvalues?

Euler was most certainly not motivated by matrices and eigenvalues, the chain of causation goes the other way. Sylvester only introduced the term "matrix" to denote an array of numbers in 1850, and ...
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7 votes

What was the motivation for the development of modern, intrinsic, differential geometry?

It begins with Riemann's lecture in Göttingen in 1854, Über die Hypothesen welche der Geometrie zu Grunde liegen". Here is an English translation: http://www.emis.de/classics/Riemann/WKCGeom.pdf As ...
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7 votes
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Did Clifford introduce the "Clifford torus", and for what purpose?

The Clifford torus was introduced by Clifford in 1873, not as embedded into $\mathbb{R}^4$ or $\mathbb{C}^2$, but first projectively and then intrinsically, by identifying the opposite sides of a flat ...
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7 votes

What makes the musical isomorphism, musical?

The $\flat$ map in music lowers the pitch of a note (by one half) and the corresponding map lowers indices, the $\sharp$ map raises it, as well as the indices. And of course adding a flat after a ...
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6 votes

What was the motivation for the development of modern, intrinsic, differential geometry?

I don't have references on hand, but I believe intrinsic geometry began with Gauss' Theorema Egregium, which asserts that the Gaussian curvature of a surface at a point depends only on the intrinsic ...
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6 votes
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Was the term "manifold" (or its German equivalent) chosen with the verb "to fold" in mind?

Notion of manifold is often motivated today by examples of simple surfaces, including developable ones, so this is a natural guess, but it is unlikely to be the case historically. German ...
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6 votes
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History of the derivative/tangent of a curve

Keep in mind that the repertoire of curves available before calculus was very limited. The earliest known definition is given for circles by Euclid in book III of Elements:"A straight line is said to ...
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6 votes
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Where can I find the original presentation of the proof, due to Grothendieck, of the $\bar\partial$-Poincaré lemma?

An alternative name is the Dolbeault-Grothendieck lemma. Dolbeault himself writes the following: "It is proved by P. Dolbeault in the $C^\omega$ case, by homotopy, as can been [sic!] the Poincare ...
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6 votes
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When was a partition of unity discovered?

Partitions of unity were formally introduced by Dieudonne (C. R. 205 (1937) 593-595), and for some time they were even called "Dieudonne decompositions". However is some special cases they were used ...
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6 votes
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Who first had the idea to study surfaces via rings of functions, as in algebraic geometry?

A canonical reference on this is Dieudonne's History of Algebraic Geometry. An abridged version Historical Development of Algebraic Geometry is freely available, see also Easton's slides. Let me make ...
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6 votes
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Who developed The Fundamental Theorem of Curves?

Existence claims as theorems became fashionable after Hilbert introduced the axiomatic method. Before that people more often talked about problems and constructions (following Euclid's, or rather ...
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5 votes
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How did the exterior product get its symbol?

Cajori gives the early uses of logical symbols in volume 2 of History of Mathematical Notations. Neither Boole nor Schröder used $\wedge$ and $\vee$ in Boolean algebra, but rather $\cdot$ (or blank) ...
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5 votes

Was the term "manifold" (or its German equivalent) chosen with the verb "to fold" in mind?

The word "Mannigfaltigkeit" itself was not invented by Riemann. It is a word meaning (great) variety or richness of types, in an every day sense. It might be used to describe that there are a variety ...
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5 votes

What did Jacobi, who lived before Riemann, have to do with the equation and theorem named after him in Riemannian geometry?

This is a natural question given the modern exposition, where Riemannian geometry is often the only place where some parts of the calculus of variations are encountered. And a reminder that historical ...
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