Timeline for Who first proved that a vanishing Riemann tensor is sufficient for the existence of Euclidean coordinates?
Current License: CC BY-SA 3.0
12 events
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| Feb 8, 2022 at 11:47 | vote | accept | Igor Khavkine | ||
| Mar 18, 2017 at 14:10 | comment | added | Igor Khavkine | @RobertBryant, thanks that's another good point. But also, following up on Francois Ziegler's comments, the existence part of what we now call Frobenius' Theorem had already appeared in an equivalent form in a paper by Deahna (1840). This history is reviews with more context in Hawkins (2005). So it seems that there was no shortage of 'traditional methods' at the time. It is a bit disappointing to see how few people bothered to write out complete proofs in that era. | |
| Mar 18, 2017 at 14:02 | comment | added | Holonomia | @IgorKhavkine: Actually, Riemann itself do claim that the vanishing is sufficient. To be precise he claimed something sharper (as consequence of the counting argument: a metric depends on $n(n+1)/2$ functions a change of coordinates depends on $n$ hence should be $n(n-1)/2$ "excesses" or "invariants" . In Spivak's volume II you can see the precise statement: it is enough that the sectional curvature vanish in $n(n-1)/2$ 2-planes to get flatness). For more see : Antonio J. Di Scala, On an assertion in Riemann's Habilitationsvortrag, Enseign. Math. (2) 47 (2001), no. 1-2, 57–63 | |
| Mar 18, 2017 at 13:40 | comment | added | Robert Bryant | @IgorKhavkine: Riemann wouldn't have needed Frobenius' Theorem to prove sufficiency. I have no idea whether this is how he would have done it, but, since he describes geodesic normal coordinates in his lecture when he defines his 'curvature measures', he might have noted that, by elementary ODE methods (existence and uniquenss for the initial value problem), one can show that the $g_{ij}$ are constant in geodesic normal coordinates when his curvature measures vanish everywhere. That would certainly have been well within 'traditional methods' since the Euler-Lagrange equations were well-known. | |
| Mar 18, 2017 at 3:16 | comment | added | Francois Ziegler | A similar (more detailed) footnote is already in Riemann's Collected Works (1892). | |
| Mar 18, 2017 at 3:02 | comment | added | Francois Ziegler | The above Riemann quote is footnoted in a 2016 edition with the remark that “these computations were first provided” in consecutive 1869 papers of Christoffel (already quoted) and Lipschitz. (And according to his biographer, Frobenius extracted mathematical ideas from both, as well as from Jacobi and Clebsch.) | |
| Mar 17, 2017 at 23:07 | comment | added | Igor Khavkine | It seems rather difficult to extract a precise sufficiency result also from Christoffel's article. His equation (9), with one set of "Christoffel symbols" set to zero, gives a linear overdetermined PDE for the desired transformation to flat coordinates, whose integrability conditions include the vanishing of the Riemann tensor. But it is far from clear to me that the existence of solutions to such a system would have been obvious at the time. It seems that the theorem of Frobenius would only be published in 1877. | |
| Mar 17, 2017 at 22:19 | history | edited | Igor Khavkine | CC BY-SA 3.0 |
Linked to English translation of Christoffel's article.
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| Mar 17, 2017 at 21:16 | comment | added | Ivan Izmestiev | I've edited my answer: Christoffel's article (in German) might be the source. | |
| Mar 17, 2017 at 21:14 | history | edited | Ivan Izmestiev | CC BY-SA 3.0 |
added 199 characters in body
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| Mar 17, 2017 at 19:29 | comment | added | Igor Khavkine | Thanks for pointing to Spivak's book. He does a fine job of guessing what the "traditional methods" of Riemann could have been. I see now that Riemann did make the claim of sufficiency. Still, I would like to know where an explicit proof was first published, even if in a monograph or a textbook. | |
| Mar 17, 2017 at 18:06 | history | answered | Ivan Izmestiev | CC BY-SA 3.0 |