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 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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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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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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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 ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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