Skip to main content

Questions tagged [differential-geometry]

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

Filter by
Sorted by
Tagged with
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. ...
Danu's user avatar
  • 3,862
28 votes
1 answer
976 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 ...
Danu's user avatar
  • 3,862
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 ...
Display Name's user avatar
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 ...
Mark Fantini's user avatar
11 votes
3 answers
930 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?
superAnnoyingUser's user avatar
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 ...
Paul Mariatte Blue's user avatar
11 votes
1 answer
2k 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 ...
Ryan Unger's user avatar
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 ...
Giuseppe's user avatar
  • 183
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). ...
David's user avatar
  • 153
9 votes
4 answers
737 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, ...
Wynne's user avatar
  • 375
9 votes
1 answer
911 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, ...
Todd Trimble's user avatar
8 votes
2 answers
900 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 ...
buckner's user avatar
  • 89
8 votes
1 answer
649 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 $\bar\partial$-...
Danu's user avatar
  • 3,862
8 votes
3 answers
706 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 ...
user avatar
7 votes
2 answers
835 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 ...
Golbez's user avatar
  • 173
7 votes
1 answer
305 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_{\...
Thomas's user avatar
  • 173
7 votes
1 answer
1k 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 ...
Danu's user avatar
  • 3,862
7 votes
0 answers
93 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.)
Archelon's user avatar
  • 163
6 votes
1 answer
162 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&:=\...
Slate's user avatar
  • 163
6 votes
1 answer
401 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 ...
Chandler's user avatar
  • 163
6 votes
2 answers
491 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 ...
helveticat's user avatar
6 votes
1 answer
193 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 ...
Ryan Unger's user avatar
6 votes
2 answers
646 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 ...
user2554's user avatar
  • 4,489
6 votes
0 answers
307 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 ...
Alan U. Kennington's user avatar
6 votes
0 answers
42 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 ...
Giuseppe's user avatar
  • 183
5 votes
3 answers
513 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 ...
tox123's user avatar
  • 1,094
5 votes
1 answer
1k 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 ...
Archelon's user avatar
  • 163
5 votes
1 answer
337 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 ...
Upc's user avatar
  • 151
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 ...
Nicola Ciccoli's user avatar
4 votes
3 answers
2k 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 ...
Avatrin's user avatar
  • 397
4 votes
2 answers
663 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 ...
user avatar
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 ...
Dmitry Grigoryev's user avatar
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 ...
Danu's user avatar
  • 3,862
4 votes
1 answer
168 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?
jq50's user avatar
  • 41
4 votes
1 answer
420 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 \...
KCd's user avatar
  • 5,647
4 votes
1 answer
211 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....
Mikhail Katz's user avatar
  • 5,842
4 votes
0 answers
102 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'...
red whisker's user avatar
3 votes
3 answers
258 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 ...
User0112358's user avatar
3 votes
1 answer
429 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.
user avatar
3 votes
1 answer
103 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 ...
Léo Vacher's user avatar
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, $$ x' = \cos (\phi) $$ $$ y' = \sin (\phi) $$...
BBSysDyn's user avatar
  • 133
3 votes
1 answer
392 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, $\nabla$ ...
user1620696's user avatar
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 $...
Babu's user avatar
  • 473
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, ...
Pádua's user avatar
  • 31
3 votes
0 answers
262 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 ...
Mikhail Katz's user avatar
  • 5,842
3 votes
0 answers
348 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 \...
Archelon's user avatar
  • 163
3 votes
0 answers
380 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.
wdlang's user avatar
  • 915
2 votes
1 answer
434 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 ...
Mark's user avatar
  • 129
2 votes
1 answer
728 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 ...
Arctic Char's user avatar
2 votes
1 answer
264 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 ...
Overflowian's user avatar