TITLE: Foliation of $\mathbb R^n$ by connected compact manifolds
QUESTION [7 upvotes]: Does there exist a smooth nontrivial fiber bundle $p: F \hookrightarrow \mathbb R^n \to B$ such that $F$ and $B$ are connected manifolds with $F$ compact? "Nontrivial" here means the fiber $F$ is not a point.

REPLY [14 votes]: On the other hand, if you only mean "foliation" as in your title, and not "fibration", then there is Vogt's foliation of R^3 by circles! (But it is not C^1, only differentiable).
Vogt, Elmar, "A foliation of R3 and other punctured 3-manifolds by circles",
Publications Mathématiques de l'IHÉS, Tome 69  (1989), p. 215-232
http://www.numdam.org/item/PMIHES_1989__69__215_0/