Infinitesimal obstructions to weakly mixing

Abstract

Let ϕ^t be the flow (parametrized with respect to arc length) of a smooth unit vector field v on a closed Riemannian manifold M ⁿ, whose orbits are geodesics. Then the (n-1)-plane field normal to v, ⊥v, is invariant under dϕ^t and, for each x ∈ M, we define a smooth real function Λ _x (t) : (1 + ⋋ _i (t)), where the ⋋_i(t) are the eigenvalues of AA ^T, A being the matrix (with respect to orthonormal bases) of the non-singular linear map dϕ^2t, restricted to ⊥v at the point ϕ_x ^-t ∈ M ⁿ.

Among other things, we prove the

Theorem (Theorem II, below). Assume v is also volume preserving and that Λ ^′'_x (t) ≥ 0 for all x ∈ M and real t; then, if ϕ ^t_x : M → M is weakly missng for some t, it is necessary that ⌈▽v⌈x ≠ 0 at all x ∈ M.