# New manifestations of the Darboux's rotation and translation fields of a surface

### 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 $S\subset\mathbb R^3$ is orthogonal to S at least at two points).

Keywords infinitesimal bending, simply connected surface, total mean curvature.

Classification (MSC2000) 53A05, 53C25.