Questions tagged [differential-geometry]
For questions about the discipline that uses differential calculus and linear algebra to study geometrical problems.
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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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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 ...
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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 ...
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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 $...
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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 ...
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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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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 ...
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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'?
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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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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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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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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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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 ...
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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 ...
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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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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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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 ...
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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 ...
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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 ...
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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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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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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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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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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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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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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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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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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 ...
Danu♦
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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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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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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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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 ...
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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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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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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 ...
CommunityBot
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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 ...
CommunityBot
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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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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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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'...
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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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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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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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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{...
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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 ...
CommunityBot
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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 ...
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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.
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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.
...
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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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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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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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