# 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.