Questions tagged [riemannian-geometry]

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17
votes
2answers
3k views

What was the relationship between Einstein and Minkowski?

I read many Einstein's Biographies, but Minkowski was never mentioned, though his discovery of the union of space and time created the basis for GR. Minkowski was Einstein's teacher of mathematics ...
8
votes
1answer
175 views

What are the references for Riemann's discussion of gravity?

I've heard anecdotally that Bernhard Riemann was interested in applying his discovery/invention of Riemannian geometry to model gravitation about 50 years before Einstein managed to do so. In this ...
9
votes
1answer
147 views

What theorem of Sophus Lie on the number of geometries is H. Poincaré referring to?

In this quotation from Henri Poincaré's essay "Non-Euclidean Geometry" published in Nature in 1892 (No. 1165, Vol 45, p. 406), he refers to a theorem of Sophus Lie. Does anyone know a source for this ...
9
votes
4answers
640 views

Help translate from German a quote by Hermann Weyl in Space Time Matter

I would like to find an accurate translation to the following quote from Space Time Matter: Man muß gegen diese Orgien des Formalismus, mit dem man heute sogar die Techniker zu belästigen beginnt, ...
3
votes
1answer
96 views

Who was first to recognize the link between (synthetic) elliptic geometry and geometry on the sphere?

Riemann was the first to talk about elliptic geometry: Bolyai and Lobačevskij (even Gauss too) studied only hyperbolic geometry. But of course some theorems of (planar) elliptic geometry were known ...
0
votes
0answers
76 views

Riemann surfaces and covering

Assuming we have a Riemann surface $S$ of degree $n$ and we look at it as a covering of the projective line $\mathbb{P}^1$. If $B$ is the set of branch points of $S$ (when $B$ is a subset in $\mathbb{...
2
votes
1answer
114 views

Material models of Riemann surfaces

It is known that during the last quarter of the 19th century there was a flourishing of the production of material models (from plaster, strings, card-board etc) of curves and surfaces in Germany (but ...
2
votes
1answer
134 views

How is “soul” meant to be understood in the context of the “Soul Theorem”

My mathematics are still quite rudimentary, but am I correct in assuming this is a reference to the "finite" state of closed manifolds as opposed to a potential, "infinite" state of the non-compacted ...
2
votes
1answer
430 views

Did Clifford introduce the “Clifford torus”, and for what purpose?

The Clifford torus shows up a lot in differential geometry in connection with minimal surfaces, for example in the Lawson's conjecture, the Oh's Conjecture, etc. It can be described as the following ...
7
votes
1answer
818 views

How did the exponential map of Riemannian geometry get its name?

I've read in several books, including Milnor's Morse Theory and Petersen's Riemannian Geometry, that the exponential map in Riemannian geometry is named so because it agrees with the exponential map ...
4
votes
3answers
374 views

Who do I blame for non-Euclidean geometry?

When, and where, was the earliest record of non-Euclidean geometry? It was always my impression that Riemann created them, but did it really take that much time for someone to try to create an ...
5
votes
1answer
235 views

What is the origin of the use of “g” for a Riemannian metric?

I am asking about the reason for the use of this letter, if known, as well as the initial occasion of its use. Ideas that have been suggested concerning the former include: That it stands for ...
3
votes
2answers
542 views

History of impact of non-Euclidean geometry on math, philosophy, and the public

I'm interested in the impact of the discovery of non-Euclidean geometry on math, philosophy, and the attitudes of the general public. I don't know anything about how things changed right after the ...
8
votes
1answer
239 views

What were Riemann's other two submissions for his habilitation?

In Stalking the Riemann Hypothesis, Rockmore discusses how Bernhard Riemann, as per custom, submitted three potential areas of research for his habilitation. Gauss was the chairman of the committee ...