9
votes
0answers
309 views

How come Cartan did not notice the close relationship between symmetric spaces and isoparametric hypersurfaces?

Elie Cartan made fundamental contributions to the theory of Lie groups and their geometrical applications. Among those, we can list the introduction of the remarkable family of Riemannian symmetric ...
23
votes
8answers
2k views

Are there some other notions of “curvature” which measure how space curves?

I am learning differential geometry and have a few questions on curvature. -- Background: Gauss invented "Gauss curvature" to measure how surface curves. Riemann gives an ingenious generalization ...
15
votes
2answers
825 views

Where did Sophus Lie write the group commutator for two one parameter groups

If $X,Y$ are vector fields and $\def\Fl{\operatorname{Fl}}\Fl^X_t$ and $\Fl^Y_t$ their local flows, let $[\Fl^X_t,\Fl^Y_t]:= \Fl^Y_{-t}\Fl^X_{-t}\Fl^Y_t\Fl^X_t$ denote the group commutator of the ...
9
votes
0answers
397 views

What is Quillen's contribution to index theorem?

In the book "Heat Kernels and Dirac Operators" by Berline, Getzler and Vergne it is said that "Our book is based on a simple principle, which we learned from D. Quillen: Dirac operators are a ...
27
votes
4answers
2k views

What, precisely, does Klein's Erlangen Program state?

People write that the Erlangen Program is a "program" (like the "Langlands Program"), i.e. a series of related conjectures, which in this case were all solved. There are various intuitive accounts, ...
8
votes
2answers
447 views

The history of the geometrization of closed surfaces

Who first recognized that the torus supports a flat structure? Who first characterized the moduli space of flat structures on the torus? Who first recognized that the closed, orientable genus 2 ...
13
votes
3answers
2k views

How to Tackle the Smooth Poincare Conjecture

The last remaining problem in this whole "everything is a sphere" business, is the Smooth Poincare Conjecture in dimension 4: If $X\simeq_\text{homo.eq.} S^4$ then $X\approx_\text{diffeo} S^4$. ...
4
votes
2answers
547 views

References for the Poincaré-Cartan forms

Hello, everybody. I'm looking for some reference about the Poincaré-Cartan form, I do not know how it is defined, I just know that it is used in Lagrangian mechanics but I have not found any ...
9
votes
3answers
919 views

History surrounding Gauss Theorema Egregium and differential geometry

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 Gauss Theorema Egregium, that is the Gaussian ...
18
votes
2answers
1k views

History of connections

Are there any good detailed historical sources about development of connections on vector/principal bundles over the last 100 years? The best source I am aware of is Michael Spivak's 5 volume opus, ...
24
votes
4answers
2k views

Why are differential forms called closed and exact?

It seems to me that "exact" relates to exact differential equation. So, why are they called exact?
5
votes
3answers
639 views

Which Bianchi identity is due to Bianchi (or not, since it might be due to Ricci (according to Levi-Civita (according to MO))) or vice versa?

wikipedia doesn't say, nor my Berger Panorama book (but I might google Levi-Civita to get rid of one level of brackets) and the library is far (actually not, but it has German Schließungszeiten and I ...
19
votes
3answers
2k views

Is “Cartan's magic formula” due to Élie or Henri?

The formula $\mathcal{L}_X\omega = i_Xd\omega + d(i_X \omega)$ is sometimes attributed to Henri Cartan (e.g. Peter Petersen; Riemannian Geometry 2nd ed.; p.380) and sometimes to his father Élie (e.g. ...
5
votes
2answers
555 views

Why did the word “exterior” get chosen for the idea of “exterior derivative”?

What are the intuitive and historical reasons for choosing the word "exterior" for the concept of an exterior derivative of a form? The reasoning I've heard about it is the following: let p(t) be a ...
3
votes
4answers
847 views

Equivalent definitions of Gaussian curvature

I'm trying to find out more about geometry of surfaces and, in particular, Gaussian curvature. I understand that it can be defined in terms of the principal curvatures (extrinsically) and also ...
2
votes
1answer
510 views

Historical question Cauchy-Crofton theorem vs. Radon transform

The Radon transform apparently was discovered around 1917 if Wikipedia is to be believed. The Cauchy-Crofton theorem is a much older theorem (mid 19th-century). But both ideas are more or less the ...