“These three structures are said to be compatible if they satisfy”

— for “r” read “w” ? WFL ]]>

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— for “r” read “w” ? WFL

]]>In fact, maybe n needs to be at least 4 to get to examples which are not known to be Andrews-Curtis trivial (but I am probably out of date on the state of the art).

]]>http://www.maths.usyd.edu.au/u/don/details.html#pubs ]]>

“A geometrical proof of Liouville’s theorem given by Cappelli [4] and reproduced by Blaschke [3] (without bothering with the assumptions to be made) is based on:

(i) Dupin’s theorem that if three surfaces of class C” cut orthogonally, then the arcs of intersection are lines of curvature on each surface, and on

(ii) the circumstance that, if every direction at every point on the surface is a direction of a line of curvature, then the surface is part of either a plane or a sphere.”

~~~

[3] W. Blaschke, Vorlesungen ueber Differentialgeometrie, vol. I, Berlin, 1930, pp. 100-102.

[4] A. Capelli, “Sulla limitata possibilità di trasformazioni conformi nello spazio,” Annali di Matematica, Ser. II, vol. 14 (1886-1887), pp. 227-237.

~~~

Quoted from:

Philip Hartman, Systems of Total Differential Equations and Liouville’s Theorem on Conformal Mappings, American Journal of Mathematics, Vol. 69, No. 2 (Apr., 1947), pp. 327-332.

~~~

[Disclaimer: I should add that I haven't read either Blaschke or Capelli, though these must be among the relevant references to the literature that Danny was searching for.]

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