All Questions
13 questions
7
votes
1
answer
232
views
Connection of principal fiber bundles — history
I wonder who was the first to discover the notion of principal fiber bundle and its connection (gauge field in the physical language). Wikipedia cites the book by Steenrod (1951). But was he the ...
- 183
5
votes
0
answers
324
views
Earliest reference for infinitesimal neighborhoods of the diagonal
Where was $I_x/I_x^2$ first introduced? (DG or AG) asks about the algebraic cotangent space. The paper First neighborhood of the diagonal and geometric distributions by Kock claims Grothendieck ...
- 10.5k
13
votes
1
answer
1k
views
Aleksandrov's proof of the second order differentiability of convex functions
Aleksandrov [A], proved a remarkable property of convex functions.
Theorem. If $f:\mathbb{R}^n\to\mathbb{R}$ is convex, then for almost every $x\in\mathbb{R}^n$ there is $Df(x)\in\mathbb{R}^n$ and ...
- 28k
14
votes
5
answers
1k
views
History of the notion of $(G,X)$-structure
I'm currently searching for sources and historical basis on the notion of $(G,X)$-structure as it appears in Thurston's work.
So far, it appears that he was the first to set it. Many mathematicans ...
- 403
16
votes
1
answer
733
views
Where was $I_x/I_x^2$ first introduced? (DG or AG)
Cotangent space appears in both differential geometry (DG) and algebraic geometry (AG).
In DG, given a smooth manifold $M$ and $x\in M$ one has an isomorphism $I_x/I_x^2 \cong T^*_xM$, where $I_x$ is ...
- 1,615
17
votes
0
answers
1k
views
Jets of sections of vector bundles expressed by symmetrized iterated covariant derivatives - who did it first?
The (non-unique) bundle isomorphism between the bundle $J^r E$ of $r$-th order jets of sections of a vector bundle $\pi:E\rightarrow M$ and the direct sum $$\bigoplus^r_{k=0}\vee^kT^*M\otimes E\...
- 7,817
4
votes
0
answers
301
views
Was this particular case of the tube formula known before Weyl and Hotelling?
The tube formula is a really nice result in differential geometry which relates the volume of the tubular neighborhood of a submanifold to its intrinsic geometry. It has been proved by Weyl in 1939 ...
- 4,123
60
votes
1
answer
6k
views
What were the main ideas and gaps in Yoichi Miyaoka's attempted proof (1988) of Fermat's Last Theorem?
Out of sheer curiosity I have been reading Stewert and Tall's "Algebraic Number Theory and Fermat's Last Theorem" (2001). As it contains various bits of history, I found out to my own shame that I was ...
- 7,127
6
votes
2
answers
1k
views
Shuffle (co-)multiplication and generalized Leibniz formula in tensor calculus
The headline already says it: Is anybody (except me, UPDATE: plus Gavrilov) aware of this formula for higher total covariant derivatives of tensor products?
It is the simplest application of the ...
- 1,068
39
votes
10
answers
4k
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 of ...
17
votes
2
answers
2k
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 ...
- 25.3k
6
votes
2
answers
3k
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 ...
- 463
3
votes
4
answers
2k
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 ...
- 315