Geometry of higher helicities

We revisit an interpretation of higher-dimensional helicities and Hopf–Novikov invariants from the point of view of the Brownian ergodic theorem. We also survey various results related to Arnold's theorem on the asymptotic Hopf invariant on three-dimensional manifolds and recent work on linking of a vector field with a foliation, the asymptotic crossing number, short path systems, and relations with the Calabi invariant.