Questions tagged [differential-geometry]

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

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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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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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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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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?
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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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2 answers
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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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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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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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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 ...
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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 ...
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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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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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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{...
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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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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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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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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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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 answer
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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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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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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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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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7 votes
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 $\bar\partial$-...
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22 votes
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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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2 answers
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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 ...
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1 answer
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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 ...
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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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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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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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9 votes
1 answer
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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 votes
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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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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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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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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4 votes
1 answer
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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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10 votes
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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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5 votes
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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 ...
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9 votes
4 answers
891 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). ...
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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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8 votes
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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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