New manifestations of the Darboux's rotation and translation fields of a surface
From NZJM
New Zealand Journal of Mathematics
Vol. 40, (2010), Pages 59-65
Victor Alexandrov
Sobolev Institute of Mathematics
Siberian Branch of the Russian Academy of Sciences,
Koptyug ave., 4,
Novosibirsk, 630090
Russia
and
Department of Physics
Novosibirsk State University,
Pirogov str., 2,
Novosibirsk, 630090,
Russia
Abstract We show how the rotation and translation fields of a surface, introduced by
Gaston Darboux, may be used to obtain short proofs of a well-known theorem
(asserting that the total mean curvature of a surface is stationary under an infinitesimal bending)
and a new theorem (asserting that every infinitesimal bending of any simply connected closed
surface 
is orthogonal to S at least at two points).
Keywords infinitesimal bending, simply connected surface, total mean curvature.
Classification (MSC2000) 53A05, 53C25.
