29 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 ...
Conifold's user avatar
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12 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,...
Gerald Edgar's user avatar
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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 ...
Alexandre Eremenko's user avatar
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.)...
Francois Ziegler's user avatar
10 votes

Non-Euclidean geometry: motivations to develop it at the times of Gauss?

The motivation of discoverers of non-Euclidean geometry (Gauss, Lobachevski and Bolyai) was their attempts to prove the Fifth postulate of Euclid (to deduce it from the other axioms, or to replace by ...
Alexandre Eremenko's user avatar
9 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 ...
Francois Ziegler's user avatar
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 ...
Conifold's user avatar
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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 ...
Conifold's user avatar
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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 ...
Conifold's user avatar
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7 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 ...
Alexandre Eremenko's user avatar
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 ...
Conifold's user avatar
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7 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 ...
Conifold's user avatar
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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 ...
Nicola Ciccoli's user avatar
7 votes
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Did Riemann invent the Riemann curvature tensor?

A short answer: yes, he did. Riemann's habilitation lecture was aimed at a broad non-mathematical audience, so he did not use formulas in it, trying to explain everything in words. The curvature ...
Alexandre Eremenko's user avatar
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 ...
Conifold's user avatar
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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 ...
Conifold's user avatar
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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 ...
Conifold's user avatar
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5 votes

Does Gauss own two “Theorema”?

In his Disquisitiones generales circa superficies curvas (1827), §12, page 24, Gauss called egregium [sponte perducit ad egregium, i.e. spontaneously leads to excellent] the following Theorem: Si ...
Mauro ALLEGRANZA's user avatar
5 votes
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Who first wrote down $S^6$'s standard almost complex structure? And who first proved that it is not integrable?

According to this arxiv paper by Atiyah, existence and construction dates from 1947; non-integrability from 1951. Here is Atiyah's history: Ehresmann 1947: Introduced the notion of almost complex ...
nwr's user avatar
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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) ...
Conifold's user avatar
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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 ...
Conifold's user avatar
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5 votes

Who do I blame for non-Euclidean geometry?

Usually this is attributed to, in alphabetical order, Bolyai, Gauss, and Lobachevski all working at about the same time in the first third of the 19th century. This certainly predates Riemann. A ...
Mikhail Katz's user avatar
  • 5,506
4 votes

The history of different constructions of tangent spaces

I apologize for the limited information in this answer, but as the question has gone unanswered for 4 years, I plead that something is better than nothing. Your fourth version appears in The ...
Michael Weiss's user avatar
4 votes

Who first had the idea to study surfaces via rings of functions, as in algebraic geometry?

The idea is usually attributed to Dedekind and Weber in Theorie der algebraischen Functionen einer Veränderlichen (1882): [1, 2, 3, 4, 5,...].
Francois Ziegler's user avatar
4 votes

History of the derivative/tangent of a curve

Carl B. Boyer's The History of the Calculus and Its Conceptual Development is especially detailed on early work relating to calculus (it pretty much fizzles out in the mid 1800s, though), and for ...
Dave L Renfro's user avatar
3 votes

Einstein already used the idea of time orientation when formulating General Relativity?

I believe the question started being studied in earnest in the 1950s, with formal definition and first results usually attributed to Markus (1955, p. 412) — by, e.g., Hawking in his famous 1966 prize ...
Francois Ziegler's user avatar
3 votes
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How did Einstein arrive at his field equations without the Bianchi identities or variational formulation?

Pais does not "suggest" that Einstein wrote the equation in this form, he reproduces on p.256 what Einstein wrote on November 25, 1915 with reference to his presentation to the Prussian Academy: $R^{\...
Conifold's user avatar
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3 votes

Who do I blame for non-Euclidean geometry?

Regarding first developments in non- euclidean geometry, Eugenio Beltrami considered Lobachevsky-Bolyai geometry as nothing else but euclidean geometry on a space with (constant) negative curvature. ...
Margaret Friedland's user avatar
3 votes

Material models of Riemann surfaces

Kharkiv University (Ukraine) subscribed to all models made M. Schilling, who probably was a student of Klein, and who run a company making and selling these models. Currently they photograph them and ...
Alexandre Eremenko's user avatar
3 votes

First appearance of Hadamard's lemma on smooth functions

The reference to the book on wave propagation (Leçons sur la propagation des ondes et les équations de l'hydrodynamique. Cours du collège de France, Hermann, 1903) available at https://archive.org/...
David Chiron's user avatar

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