TITLE: Does every open manifold admit a function without critical point?
QUESTION [7 upvotes]: Assume that $M$ is  a non compact smooth manifold. Is there a smooth map $f:M\to \mathbb{R}$ such that $f$ has no critical point?
The motivation comes from the conversations on this post.

REPLY [11 votes]: Interesting question. I did a little research, and found out that apparently the answer is yes: A 1961 paper by Morris Hirsch (Theorem 4.8) showed that every noncompact (connected) smooth manifold admits a smooth real-valued function with no critical points.