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Questions tagged [differential-geometry]

For questions about the discipline that uses differential calculus and linear algebra to study geometrical problems.

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3
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1answer
94 views

From where the so-named “elastica problem” is coming from?

In a book by Cash et al, I see the mention of the so-called Elastica problem (pg 221 in the link here). The problem is presented as a system of ODEs, $$ x' = \cos (\phi) $$ $$ y' = \sin (\phi) $$...
3
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0answers
51 views

Asymptotically Periodic Potentials

Who came up with the idea of solving elliptic equations with periodic potentials and from there solving elliptic equations with asymptotically periodic potentials? I heard it was Pierre Louis Lions, ...
6
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1answer
210 views

Who developed The Fundamental Theorem of Curves

In any modern differential geometry textbook (Do Carmo, for example), the fundamental theorem of curves can be found. It states that: every regular curve in three-dimensional space, with non-zero ...
9
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4answers
638 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
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0answers
149 views

Priority on lemniscate of Gerono?

The Lemniscate of Gerono is a special case of the Lissajous curves. The dates for the two mathematicians are fairly close: Gerono (1799-1891) and Lissajous (1822-1880). Historically who has priority ...
1
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1answer
64 views

Summary of Gauss's work on geodesic lines on ellipsoid

The solution to the problem of geodesic lines on a biaxial ellipsoid (when two of the axes are equal) is not very hard and can be solved by mathematical tools that existed prior to Gauss - i.e via ...
3
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1answer
197 views

Why is distance sometimes abbreviated S?

While distance in physical formulas is often abbreviated as d (which is pretty intuitive), another common abbreviation is s, as seen e.g. here, here or here. It also seems to be used in optics to ...
3
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1answer
485 views

What is the origin of French/Burmester's curves?

French curves are a set of curvilinear rulers used in industrial design, before the advent of CAD, when everything still had to be drawn by hands. The most popular set of such rulers is made up of 3 ...
2
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1answer
83 views

Nomizu's structural approach to differential geometry

In this article in Wikipedia about Katsumi Nomizu https://en.wikipedia.org/wiki/Katsumi_Nomizu it is written that "Over the course of his career, Katsumi Nomizu was influential in determining the ...
0
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0answers
54 views

How to derive from Gauss's result on the volume of orthoscheme tetrahedron the formulas of Lobachevsky and Bolay?

My question is a direct continuation of my already posted question Did Gauss's expression for the differential of the hyperbolic volume of the tetrahedron agree with later results?. I simply didn'...
4
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1answer
219 views

First appearance of Hadamard's lemma on smooth functions

Hadamard's lemma, in one dimension, says for any smooth function $f \colon \mathbf R \rightarrow \mathbf R$ there is a first-order expansion of $f$ at $0$: $f(x) = f(0) + xg(x)$ where $g \colon \...
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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
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1answer
113 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
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3answers
172 views

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

This idea provides the foundations of algebraic geometry now; and they have certainly gone down the rabbit hole with it. As a student studying this subject, I have always found it such a great leap to ...
2
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1answer
241 views

When was a partition of unity discovered?

A partition of unity is a mathematical concept in geometry. I want to know when and in what context this concept appeared.
4
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1answer
156 views

Who first wrote down $S^6$'s standard almost complex structure? And who first proved that it is not integrable?

It is well known that $S^6$ admits an almost complex structure, inherited from its manifestation as the space of unit imaginary octonions. This almost complex structure is also well-known not to be ...
4
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1answer
180 views

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

The theory of General Relativity as usually presented currently defines the relativistic spacetime as a tuple $(M,g,\nabla, T)$ where $(M,g)$ is a four dimensional smooth lorentzian manifold, $\nabla$ ...
2
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1answer
422 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 ...
4
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1answer
187 views

Does Gauss own two “Theorema”?

When I read our differential geometry book, I saw two theorema: "Theorema Egregium" and "Theorema Elegantissimum". Mathematically, they are not the same. On wikipedia, there is nothing about ...
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0answers
84 views

Gauss fundamental form in differential geometry : use of dot products

In textbooks on differential geometry, the first fundamental form looks like $E^2+2FG+H^2$, and its length is calculated through the help of the dot product. However, the inner product did not exist. ...
7
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1answer
180 views

How did Einstein arrive at his field equations without the Bianchi identities or variational formulation?

When we introduce the Einstein equations in courses on General Relativity we use either the Bianchi indentity or the the variational principle to motivate the appearance of the Einstein tensor $$ G_{\...
5
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1answer
111 views

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

In Riemannian geometry we have two very important things named after Jacobi: the Jacobi equation $J''=R(\gamma',J)\gamma'$ and Jacobi's theorem which states geodesics never minimize past conjugate ...
5
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1answer
247 views

Where can I find the original presentation of the proof, due to Grothendieck, of the $\bar\partial$-Poincaré lemma?

In complex geometry, there is the a lemma, analogous to the Poincaré lemma in (real) differential geometry, which states that a $(p,q)$-form that is $\bar\partial$-closed is locally $\bar\partial$-...
19
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1answer
2k views

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

General Relativity was developed with Differential Geometry as the tool. How was Einstein led to make a contact with Differential Geometry for his theory of General Relativity? Who suggested him to ...
0
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2answers
183 views

why was the hairy ball theorem important

In Topology courses one learns An even dimensional sphere does not possess any continuous field of unit vectors What is the importance of this result? I can't think of any applications off the top ...
0
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1answer
93 views

on the classification of singular points

After reading this question and the answers to it, I am interested o know who were the first mathematicians who started classifying singular points of curves: i.e. different kind of nodes, of cusps ...
4
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3answers
721 views

History of the derivative/tangent of a curve

I just want to know the history of the derivative. Whenever I Google for it, I find the history of calculus or the tangent of a curve. However, they barely touch upon what happened before Leibniz and ...
5
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0answers
80 views

Whence originates the use of the nabla (∇) for a connection or covariant derivative?

Who introduced it, when, where, and with what if any rationale? (Note that I am not asking about the origin of the nabla symbol, which is covered here.)
7
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3answers
774 views

Who wrote down the first differential equation?

I am just curious who was the first person to write down a differential equation? And what was this differential equation?
5
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1answer
243 views

How did the exterior product get its symbol?

As per the title: where did the notation $a\wedge b$ for the exterior product of $a$ and $b$ originate, and/or who popularised it? I'm especially interested in motivation for the choice of this symbol ...
7
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1answer
817 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
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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 ...
2
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0answers
124 views

Why is the space of sections of $E$ called $\Gamma(E)$?

The space of sections of a bundle $\pi: E \to B$ is commonly denoted $\Gamma(E)$. (Note that the graph of a function $f$ is $\Gamma(f):=\left\{(x,f(x))\right\}$, and a particular section $\sigma: B \...
5
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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 ...
1
vote
1answer
181 views

Debauches of indices: Translation request

Finally I found the source of the dictum "debauches of indices". It is most often used in singular, as in Spivaks's Vol.II p.211. The original is from the first preface in E. Cartan's "Lecons sur la ...
3
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1answer
121 views

Did anybody consider the product of the principal curvatures before Gauss?

Gauss proved that the so-called Gaussian curvature is an intrinsic invariant of the surface, even though it is defined extrinsically as the product of the principal curvatures of a Euclidean embedding....
8
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1answer
745 views

How was curvature originally defined and calculated?

I am interested in the early history of curvature. Who defined it first and when, who came up with the name, how was it calculated before mathematicians used calculus to define $k=|α''(s)|$? Are there ...
5
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0answers
212 views

Is $\Gamma^i_{jk}$ the Christoffel symbol or the Christoffel symbols?

For years, I have been perplexed that the expression $\Gamma^i_{jk}$ is often referred to in the plural as "the Christoffel symbols", although sometimes it is referred to in the singular as "the ...
8
votes
4answers
528 views

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

Recently I came across several papers of Monge and Lagrange, around the end of the 18th century, considering developable surface as 'folded' planes, using specifically the word "plié" (i.e. folded). ...
5
votes
1answer
265 views

Did Gauss formulate, or at least know of, the full essence of the Gauss-Bonnet Theorem?

I know that a special case of the Bonnet theorem, called the Theorema Elegantissimum, was proved by Gauss in his 1827 treatise on differential geometry. This was a theroem that dealt with the ...
7
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2answers
381 views

Where did Cartan introduce his notation for basis vectors and covectors?

There is a notation used in differential geometry and general relativity in which the partial derivative operators $\partial_\mu$ are used as the basis for the space of contravariant vectors, and ...
6
votes
2answers
490 views

The origin of the name “connection” in differential geometry

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 ...
2
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0answers
316 views

How was the use of upper and lower indices in tensor notation developed?

It is just a notation, but it is so economical and so systematic. So who invented them? A handy notation should be helpful for the development of the whole field.
2
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0answers
85 views

Motivation behind Euler Theorem in differential geometry [duplicate]

I am teaching a class on elementary differential geometry and I would like to know, for myself and for my students, something more about the history of Euler Theorem and Euler equation: the curvature ...
6
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0answers
34 views

Motivation behind Euler Theorem in differential geometry [duplicate]

I am teaching a class on elementary differential geometry and I would like to know, for myself and for my students, something more about the history of Euler Theorem and Euler equation: the curvature ...
9
votes
2answers
993 views

Motivation behind Euler Theorem in differential geometry

I am teaching a class on elementary differential geometry and I would like to know, for myself and for my students, something more about the history of Euler Theorem and Euler equation: the curvature ...
7
votes
1answer
310 views

How did the 'Poincaré patches' get their name?

De Sitter space and Anti de Sitter space are two of the most important solutions to the Einstein field equations. One famous method to obtain these spacetimes is to consider a $N$-dimensional ...
18
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3answers
1k views

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

I know that tensor calculus was developed around the same time as general relativity. Tensor calculus was the prime way to deal with geometric objects, based on expliciting all coordinates and doing ...
28
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1answer
600 views

The history of different constructions of tangent spaces

In Lee's book 'Introduction to Smooth Manifolds', there is an interesting discussion (near the end of chapter three) of several different ways of viewing/constructing the notion of a tangent space to ...
10
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3answers
601 views

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

How come do we attribute general theory of relativity to Einstein when David Hilbert published first?