Around problem 8.1: Augmenting an injective immersion into a diffeomorphism

Abstract

This chapter addresses the question of the extension of an injective immersion into a diffeomorphism. This is done by complementing continuously the full-rank rectangular Jacobian of the injective immersion into an invertible square matrix. Indeed, when this is possible, an explicit formula for the diffeomorphism is given. Several sufficient conditions for a Jacobian complementation are given with either explicit formulas or constructive algorithms and are illustrated in the example of an oscillator with unknown frequency.