Closed integral curves in 3-space and isotopic two-dimensional deformations

Abstract

If there is given an isotopic of a two-dimensional euclidean plane E onto itself, one can adjoin to every point P of E a certain closed curve, the so-called of P (?1). If the indicatrix does not pass through P, we introduce the order of P relative to its indicatrix as the of the at P. A relation between the rotation numbers in different points of E (?3) and a formula for the rotation number in the general case of a bounded deformation (?4) is established. This formula admits an application to the problem of closed integral curves of continuous vector fields in the 3-dimensional sphere.