Variations and deformations

Abstract

In this chapter several problems concerning variations and deformations are dealt with. The general form of such a problem is as follows: certain geometrical objects suffer a finite or infinitesimal transformation and the behaviour of some other objects depending on them is required. In deformation problems we deal with the special case where the variation is due to displacements of some kind, for instance a dragging along or a parallel displacement. The case occurring most frequently is that some objects are left at rest, others are dragged along and others are displaced parallel. The theory of variation and deformation is very important because a great number of problems in differential geometry can be treated in a very elegant way by using the methods of this theory.