Critical points and surjectivity of smooth maps

Abstract

Let f:Mm→Nn be a smooth map between two differential manifolds with N connected, f(M) closed and f(M)≠N. In this short note, we show that either all the points of M are critical points of f or the dimension the collection of all critical points of f is not less than n−1. Some consequences of this result for surjectivity of mappings are also presented.