PARALLELIZABLE FLOWS AND LYAPUNOV'S SECOND METHOD*

This paper is divided into two parts. Part I deals with flows on arbitrary metric spaces and answers completely the question when they are parallelizable. We give an elementary proof for arbitrary locally compact separable metric spaces which, incidentally, also clarifies the role of Niemytskii's notion of an improper saddle point. Although our main result can be established very simply in special cases, by direct appeal to Feldbau's theorem [8], e.g., when the underlying space is an ANR, it does not appear to be so easily accessible in the general case considered here. In Part II we relate the geometry of parallelizable flows to questions concerning the existence of Lyapunov functions. Although we consider only the important particular case of flows on En x E determined by the trajectories of differential equations (and so do not need the full generality of Part I), our results extend verbatim to the case of flows discussed in I. We find, with the aid of suitable sections, that even when there is no stability at all there always exist functions with two of the three properties required of Lyapunov functions, and that the regularity properties of these functions depend essentially upon those of the sections. We carry out all constructions on the rectified flow. This affords conceptual simplification and, more important, results in a unified approach which brings out clearly the geometric picture that has been obscured, if not totally lost, in previous constructions (cf., e.g., [3], [4]). As illustration of our method we give simple proofs of the converses of the classical Lyapunov theorems.