An example of a local flow on a manifold

Abstract

Let p be a point of a smooth n-dimensional manifold. If n is even it is easy to construct a local flow about p such that p is an isolated critical point and no orbit except the stationary one at p has p as a limit point. We call such a flow a nonnull flow about p (NN-flow). Mendelson has conjectured that NN-flows do not exist on odd dimensional manifolds. We show that Mendelson’s conjecture is false by constructing an NN-flow on any smooth manifold whose dimension is an odd integer exceeding one.