Questions tagged [differential-geometry]
For questions about the discipline that uses differential calculus and linear algebra to study geometrical problems.
56
questions
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Did the formation of Differential Geometry come before Differential Topology/ Topology in general?
I’m pretty interested in the history of mathematics, and it has always been my belief that the great pioneers of Differential Geometry were Gauss and Riemann, and the father of topology was mostly ...
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0answers
87 views
historical comments in bad taste
On page 44 of [1] Flanders writes "A treatment from this point of view of exterior calculus
which is not quite completely satisfactory and which unfortunately is embellished
with historical ...
3
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1answer
109 views
Complete list of publications of Rebecca Barlow
Rebecca Barlow is the discoverer of an interesting surface in algebraic geometry. Is anybody aware of a full list of her contributions? Has she continued working in mathematics in the 21st century?
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1answer
372 views
What is the etymology of the mathematical terms "sheaf, stalk, germ"?
The peculiar agricultural terminology commonly used in algebraic geometry and category theory, "sheaf", "stalk", "germ", is quite well-known. A sheaf is pictured as something like a bundle of stalks, ...
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38 views
Were Christoffel symbols used by Riemann?
The answer to the question would appear to be "no" looking at the transcription of Riemann's 1854 Ueber die Hypothesen, welche der Geometrie zu Grunde liegen lecture. However, Bernhard Riemann (1826-...
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2answers
576 views
What is the history of staircase or 𝜋=4 paradox?
The staircase 'paradox' has been discussed here and elsewhere a few times (search for staircase + paradox).
My question is whether this puzzle has been discussed in the academic literature or ...
3
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1answer
109 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
53 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
320 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
...
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4answers
686 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
230 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 ...
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1answer
87 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 ...
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1answer
722 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 ...
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1answer
2k 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
146 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 ...
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0answers
73 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
371 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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94 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
126 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
215 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
325 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
190 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 ...
3
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1answer
296 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
632 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
263 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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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. ...
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1answer
244 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_{\...
6
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1answer
143 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 ...
7
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1answer
460 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 $\bar\partial$-...
20
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1answer
3k 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
228 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
108 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
1k 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
86 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.)
6
votes
2answers
363 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 ...
8
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1answer
1k 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
430 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 ...
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0answers
232 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
812 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 ...
0
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1answer
315 views
The debauch of indices: translation request
Finally I found the source of the dictum "debauches of indices". It is most often used in singular ""debauch", as in Spivaks's Vol.II p.211. The original is from the first ...
3
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1answer
144 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....
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1answer
1k 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 ...
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257 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
838 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). ...
6
votes
2answers
469 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 theorem that dealt with the ...
8
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3answers
559 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
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2answers
717 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 ...
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337 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.
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0answers
89 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
38 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 ...