Questions tagged [differential-geometry]
For questions about the discipline that uses differential calculus and linear algebra to study geometrical problems.
69
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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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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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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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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 ...
CommunityBot
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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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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 ...
CommunityBot
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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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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 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 ...
CommunityBot
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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 \...
Danu♦
- 3,812
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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 ...
- 73.6k
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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....
CommunityBot
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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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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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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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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 ...
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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 ...
Danu♦
- 3,812
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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 ...
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