Questions tagged [differential-geometry]

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

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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
1 vote
1 answer
152 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 ...
pokssin's user avatar
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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 ...
pokssin's user avatar
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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 ...
pokssin's user avatar
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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 $...
tryst with freedom's user avatar
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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 ...
dimachaerus's user avatar
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1 answer
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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 ...
Adam Herbst's user avatar
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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'?
Apoorv Potnis's user avatar
2 votes
1 answer
421 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
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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&:=\...
Slate's user avatar
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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 ...
GaryW's user avatar
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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 ...
Chris's user avatar
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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 ...
hyportnex's user avatar
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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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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
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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, ...
Todd Trimble's user avatar
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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-...
Antoni Parellada's user avatar
8 votes
2 answers
683 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
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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, $$ x' = \cos (\phi) $$ $$ y' = \sin (\phi) $$...
BBSysDyn's user avatar
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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, ...
Pádua's user avatar
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1 answer
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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 ...
Chandler's user avatar
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4 answers
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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, ...
Wynne's user avatar
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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 ...
Mikhail Katz's user avatar
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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 ...
user2554's user avatar
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4 votes
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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 ...
Dmitry Grigoryev's user avatar
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
2 votes
1 answer
219 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
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85 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'...
user2554's user avatar
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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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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{...
David's user avatar
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2 votes
1 answer
136 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 ...
David's user avatar
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3 votes
3 answers
251 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
398 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
4 votes
1 answer
234 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
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3 votes
1 answer
377 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
2 votes
1 answer
709 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
5 votes
1 answer
324 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
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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. ...
jan roelens's user avatar
7 votes
1 answer
297 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
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6 votes
1 answer
186 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
7 votes
1 answer
599 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
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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
0 votes
2 answers
258 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 ...
john mangual's user avatar
0 votes
1 answer
113 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 ...
David's user avatar
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4 votes
3 answers
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 ...
Avatrin's user avatar
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6 votes
0 answers
91 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
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6 votes
2 answers
471 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
10 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
5 votes
3 answers
505 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
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