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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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2 votes
1 answer
273 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 ...
3 votes
1 answer
104 views

Who discovered that the electromagnetic tensor is the curvature of a connection?

I can not identify clearly who was the first one to realize that the electromagnetic tensor is the curvature 2-form of a U(1)-connection. Looking at Weyl's work, it seems that he came pretty close to ...
4 votes
0 answers
114 views

History of Pfaff's problem and integrability conditions

I'm curious about Pfaff's problem and all the various mathematicians who have worked on it. From Wikipedia articles I was able to gather the following timeline of events. In the modern language: Pfaff'...
1 vote
1 answer
173 views

What is Metric tensor's origin?

I was usually interested in metric tensor, So I have hard searched it, But most of what I was looking for was about the 'theory of relativity'. Even so, I can find information about mathematical ...
6 votes
2 answers
657 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 ...
0 votes
1 answer
107 views

What is Minkowski problem?

I have a question about 'Minkowski problem' related to Gaussian curvature. I searched 'Minkowski problem' in Google, Almost all of them were related to Einstein's theory of relativity. So I ask about ...
1 vote
0 answers
136 views

Origin of the Christoffel symbol

I am interested in the Christoffel symbol. So I found Christoffel's 1869 paper and The Road to Reality by Roger Penrose. These days, the Christoffel symbol has been established as a covariant ...
3 votes
0 answers
135 views

What motivated Grassman to study Grassman algebras in 1844?

These are $n$-tuples of reals, added componentwise and multiplied via the “exterior product.” They were introduced by Grassmann in 1844 as part of a brilliant attempt to construct a vector algebra in $...
0 votes
0 answers
119 views

Who discovered intrinsic curvature first?

Historically, Gauss measured it by an old method called after his name today: The Gauss map. This maps the normal vector at each point of a surface curve to the unit sphere. Then he measured the angle ...
10 votes
2 answers
2k views

Was Euler's theorem in differential geometry motivated by matrices and eigenvalues?

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 ...
1 vote
1 answer
235 views

Did Riemann invent the Riemann curvature tensor?

I'm pretty sure they weren't using tensors in the modern sense at that point, but to what extent did Riemann lay out the structure or significance underlying his eponymous tensor? In his habilitation ...
0 votes
0 answers
113 views

Why is bachelors' unknotting called as such and who discovered it?

Bachelors' unknotting is a way to show that all tame knots are isotopic to the unknot, by tightening a knot to a point. Why is it called 'bachelors' unknotting'?
2 votes
1 answer
437 views

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

I'm making a historical research on the origins of differential geometry, starting with non-Euclidean geometry introduced by Gauss. Reading different historical accounts, what I don't understand is ...
11 votes
2 answers
2k 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 ...
6 votes
1 answer
175 views

Why are the symbols E, F, G, L, M, and N used for the coefficients of the fundamental forms?

In differential geometry, if $e_1$ and $e_2$ are bases for a tangent space $T_pM$, then the coefficients of the first fundamental form is: $$\begin{align}E&:=\left<e_1,e_1\right>\\F&:=\...
2 votes
0 answers
103 views

How to give a meaningful interpretation to Gauss's notion of "oriented area" of self-intersecting geodesic polygons?

Gauss's notion of "oriented area" of figures characterizes the notion of two-dimensional content in a way that enables self-intersecting geodesic polygons (the term "geodesic" ...
0 votes
0 answers
85 views

What is the historical significance of Riemann’s discovery of non-flat metrics?

From a perspective of math history, what discoveries resulted directly from Riemann’s discovery and codification of non-flat metrics on Riemannian manifolds? After Albert Einstein modeled the Universe ...
4 votes
2 answers
679 views

What makes the musical isomorphism, musical?

From what I understand musical isomorphism is the formal way to speak about raising the index of a vector field (refer here). But what I don't understand is why is it called musical isomorphism! In ...
1 vote
1 answer
204 views

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 ...
6 votes
1 answer
405 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 ...
0 votes
1 answer
454 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 ...
0 votes
0 answers
101 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 ...
8 votes
3 answers
708 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
2 answers
498 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 ...
4 votes
1 answer
169 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?
9 votes
4 answers
738 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, ...
9 votes
1 answer
929 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, ...
0 votes
0 answers
61 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-...
8 votes
2 answers
908 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 votes
1 answer
119 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 votes
0 answers
54 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, ...
24 votes
1 answer
4k 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 ...
1 vote
0 answers
97 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. ...
5 votes
1 answer
423 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 \...
20 votes
3 answers
2k 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 votes
1 answer
989 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 ...
3 votes
0 answers
263 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 vote
1 answer
121 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 ...
4 votes
1 answer
4k 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 ...
5 votes
1 answer
3k 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 ...
3 votes
1 answer
395 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, $\nabla$ ...
0 votes
0 answers
87 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'...
2 votes
1 answer
140 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 ...
10 votes
4 answers
1k 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
3 answers
514 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 ...
0 votes
0 answers
116 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{...
3 votes
3 answers
261 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 ...
4 votes
1 answer
251 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 votes
1 answer
432 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.
30 votes
2 answers
2k views

When and how was the geometric understanding of gauge theories developed?

In theoretical physics, the modern perspective on gauge theory is that it is most elegantly described in the 'language' of differential geometry. I am interested in the history behind these ideas. ...