Questions tagged [differential-geometry]
For questions about the discipline that uses differential calculus and linear algebra to study geometrical problems.
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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♦
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votes
1
answer
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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♦
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votes
1
answer
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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 ...
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3
answers
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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 ...
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2
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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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3
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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?
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2
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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 ...
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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). ...
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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 ...
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4
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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, ...
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9
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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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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 ...
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3
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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 ...
user466
7
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1
answer
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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♦
- 3,812
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1
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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_{\...
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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♦
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6
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2
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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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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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6
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2
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457
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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 ...
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1
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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 ...
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2
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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 ...
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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.)
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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 ...
6
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0
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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 ...
- 183
5
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3
answers
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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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1
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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&:=\...
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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 ...
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1
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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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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 ...
- 1,707
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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 ...
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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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1
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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♦
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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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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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3
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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 ...
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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 ...
user12582
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1
answer
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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.
user5146
3
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1
answer
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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,
$$
x' = \cos (\phi)
$$
$$
y' = \sin (\phi)
$$...
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1
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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, $\nabla$ ...
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1
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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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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, ...
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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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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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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 ...
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1
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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 ...
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1
answer
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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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2
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1
answer
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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 ...
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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 $...
2
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0
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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" ...
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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 \...
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