Transversality in space of three dimensions

Abstract

So-called transversality relations arose first in the calculus of variations and later in the theory of infinitesimal contact transformations. For the plane such relations are of arbitrary character, but in all spaces of more than two dimensions only correspondences of certain specific types can be identified with transversality relations. Similarly, systems of extremals, which are of arbitrary character for the simplest problem of the calculus of variations, are of peculiar geometric character for all higher problems.t In this paper we find a simple geometric criterion for testing when a given correspondence between surface elements and line elements in three-space is of the transversality type. Briefly stated, a certain induced homography must be involutorial. This is both necessary and sufficient. The result applies to simple integrals