Question Given a manifold $M$ equipped with a vector field $\xi$ which has a description in local coordinates. Is it possible to construct a topological invariant which can be used to tell if another manifold equipped with a vector field is equivalent up to coordinate transformation.
This question can be equivalently formulated through differential $\alpha_i$ forms where $\alpha_i = \ker \xi$
Motivation Given a ordinary differential equation $\mathcal{E}$ it can be considered as a manifold equipped with the cartan distribution \begin{equation} D = \partial_{x} + p_1 \partial_{p_0} + \ldots + p_{k-1}\partial_{p_{k-2}} + F(x,p_0,\ldots,p_{k-1})\partial_{p_{k-1}} \end{equation} Having found equivalence classes of ordinary differential equations of a fixed order and its solution. I want to see if its possible to determine if a given ODE is equivalent to my ODE.
The kernel of $\xi$ is the kernel of the trace $ \text{Tr}\ \ \nabla_i v_k (x)==0$, that is the linear space of vector fields $v$ which can be transformed locally to a constant vector field, that has no sources or sinks if integrated over a small n-1 sphere.
So, by physical reasoning of fluid mechanics of incompressible fluids, such vector fields can be deformed and decomposed to a basis of flows that are in a 1-1 map with minimal paths surounding the pillars in a system of caves.
Central in such construction of cohomology classes is the invariant trace $\text{Tr} \ \xi_i^\star x_k$ under locally unitary transformations so that the integral of the trace can be used to count the number of holes.
In dealing with vector fields with addtional algebraic structures, any field of algebraic field theory generates cohomology classes, the best known in mathematical physics are n-forms (exterior algebra), spin structures (Athiyah-Singer Theorem) and BRST cohomology as the only way to interpret field calculations with Feynman rules applied to von Neumann algebra valued operators in Hilbert spaces.