Theorems similar to Tischler fibering theorem 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
$$  \text{Existence of some diffential form/s} \implies \text{Topological consequences on $M$}$$
 A: If a compact smooth manifold has a closed, non-degenerate $2$-form, i.e. it is a symplectic manifold, then it must be even dimensional and all the even Betti numbers $b_{2k}(M) = \dim(H_{2k}(M,\mathbb{R}))$ are non-zero. 
To prove it is even dimensional is just linear algebra, a vector space with a non-degenerate bilinear form must be even dimensional. To prove the fact about the Betti numbers, note that if a manifold $M$ of dimension $2n$ has a closed non-degenerate $2$-form $\omega$ then $[\omega]^{k} \neq 0 \in H^{2k}(M,\mathbb{R})$ (for $1 \leq k \leq n$), since $[\omega]^{n}$  is the class of a volume form on $M$.
This is not really considered a theorem, but I thought it was worth mentioning since the hypothesis are very similar to Tischler theorem only for forms of one dimension greater.
A: 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. 
