# Tag Info

## Hot answers tagged differential-geometry

24

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 all of them. To throw out geometry and keep laws is equivalent to describing thoughts without words. We must search for words before we can express thoughts. ...

18

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's work on manifolds with a metric, and summarized in their comprehensive 1900 book. Einstein learned about it from a geometer friend Grossman around 1912, and ...

17

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 vector potential for the magnetic field, which can be interpreted as a connection form, making magnetic theory the first gauge theory in hindsight. In the famous ...

17

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 volume 5 of Atiyah's collected works (on gauge theories) contains some historical comments on this from the mathematics side. You can read it here. This is ...

15

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 to find the correct mathematical formulation of his theory, finally discovering the necessary tools from differential geometry during a collaboration with the ...

11

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 needed to find out when it was actually discovered by each person, and we can never be 100% sure of the result. Perhaps Gauss was the first. But the usual way to ...

11

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 property of a curve, and his "calculations" are constructions of segments. The first person to "see" curvature was Oresme (c. 1320-1382), ...

11

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.)To your questions: 1) Techniker is for technicians, engineers, graduates of the Technische Hochschule where Weyl gave these lectures, as opposed to scientists ...

9

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 of Relativity (GR) in a 1907 paper "Relativitätsprinzip und die aus demselben gezogenen Folgerungen" (On the Relativity Principle and the Conclusions Drawn from ...

9

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 decades, Alexes Clairaut studied it for one-dimensional curves in his 1731 book Recherces sur les courbes à double courbure. Euler was the first to apply this ...

8

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 Geometry" (1854). Originally, "g" was for gravity. Einstein and Marcel Grossman, his mathematician friend who introduced him to tensor calculus and collaborated ...

8

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 due to Leibniz, which he used to argue against the actual existence of indivisibles, see The Philosophical Assumptions Underlying Leibniz's Use of the Diagonal ...

7

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 ...

7

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 did not do much with them. Only Cayley discovered in 1857 that each matrix satisfies equation of its own order, which led to "characteristic values". However, ...

7

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 Paul Siegel said, Gauss theory of surfaces and Lobachevski geometry gave some motivation. As another motivation, one can mention Riemann's work on in ...

7

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 parallelogram, a commonplace method in modern topology textbooks. The embedding into $\mathbb{R}^4$ first appeared in Killing's Die Nichteuklidischen ...

7

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 arcus $\mathrm{AM}$ absolvitur. Solutio. Sit spatium $\mathrm{AM}$, sive sit linea recta sive curva, $=s$, et celeritas, quam corpus habet in $\mathrm M$ sit $c$ (....

6

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 metric in a neighborhood of that point and not on the embedding of the surface (in other words, it is invariant under local isometry). This is a bit surprising ...

6

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 derive were originally derived without it. There is no "exponential map" in Gauss's General Investigations of Curved Surfaces (1825,27) or Riemann's On the ...

6

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 touch a circle which, meeting the circle and being produced, does not cut the circle". In other words, the tangent is a line that meets a curve but stays "on ...

6

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 lemma. H. Cartan brings the proof to the $C^\omega$ case by a potential theoritical method [Do 53]. Simultanously, the lemma has been proved by A. ...

6

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 by Whitney (TAMS, 36 (1934) 63-89). This information is taken from Hormander, Analysis of Partial differential operators, vol. I, comments to Chapter I.

6

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 a general comment first. When we wonder "however did someone first connect these two [modern ideas]?" we tacitly presuppose that they were always ...

6

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 Pappus's, distinction). In this case, the construction was of a curve given its curvature and torsion as functions of arclength (the "natural equations"). For the ...

5

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) and $+$, the idea was to make it as "algebra looking" as possible. Peirce in his 1865 lectures was apparently first to give Boolean algebra its modern form, by ...

5

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 Mannigfaltigkeit is associated not so much with geometric folding as with "many folded into one", alternatively translated as "multiplicity". Cantor even used ...

5

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 order rarely follows the order of modern expositions. Jacobi's equation, condition, fields, etc., are not specific to Riemannian geometry, they relate to the ...

5

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 structure and showed that the 6-sphere admits an almost complex structure, but explicitly points out that he does not know whether it has a complex structure. ...

4

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 Mathematical Theory of Relativity by Eddington, in a section titled "The mathematical notion of a vector". Eddington writes: We have a set of four numbers $(A_1,A_2,$...

4

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 of different things of a related type, like many different trees, or something along these lines. While I suppose there is an etymological connection, in the ...

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