The Picard theorem I am considereing is the one that states, under suitable hypothesis, that an ordinary differential equation has a unique solution in a suitable interval $I\subset\mathbb{R}$.
Does someone know applications of this theorem in differential geometry and differential topology?
One basic application is that any vector field can be integrated i.e given any vector field $V$ on a manifold $X$, and any point $p\in M$, there exists a path $\varphi : (-\varepsilon,\varepsilon) \rightarrow M$ with $\varphi(0)=p$ such that $$\frac{d\varphi}{dt} = V(\varphi(t)).$$ A generalization of this is the Frobenius' theorem which also needs Picard's theorem. As it happens in many cases, the idea behind the proof i.e contraction mapping, has many more applications in geometry, than Picard's theorem itself.
Would have like to post this as a comment, but I dont have enough reputation!