TITLE: Theorems similar to Tischler fibering theorem
QUESTION [5 upvotes]: Tischler theorem states that the existence of a nowhere vanishing closed $1$-form in a compact manifold $M$ implies that the manifold fibers over $S^1$. Do you know any other diffential topology results of this kind? By this kind I mean
$$  \text{Existence of some diffential form/s} \implies \text{Topological consequences on $M$}$$

REPLY [4 votes]: On a $2k+1$-dimensional manifold, a 1-form $\alpha$ such
that $\alpha\wedge (d\alpha)^k \neq 0$ at each point gives
a contact structure. I believe that it is still open
which $2k+1$-manifolds for $k>1$ admit a contact structure (
all 3-manifolds have a contact structure). 
On a 3-manifold, a nowhere vanishing $1-$form $\alpha$ such
that $\alpha\wedge d\alpha=0$ is equivalent to having a foliation
($ker\alpha$ gives an integrable plane field by the Frobenius
theorem).
This does not give much topological information, since every
3-manifold admits a (2-dimensional smooth orientable) foliation. However, if there is also
a closed 2-form $\omega$ such that $\omega \wedge \alpha >0$
(i.e. nowhere vanishing), then the foliation is taut. This
is a non-trivial topological condition, as many 3-manifolds
do not admit a taut foliation, and holds for fibered 3-manifolds,
so strictly generalizes Tischler's theorem in the 3-dimensional case in some sense.
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