Infinitesimal obstructions to weakly mixing

Let ^t be the flow (parametrized with respect to arc length) of a smooth unit vector field v on a closed Riemannian manifold Mⁿ, whose orbits are geodesics. Then the (n-1)-plane field normal to v, v, is invariant under d^t and, for each x M, we define a smooth real function _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 d^2t, restricted to v at the point _x^-t Mⁿ.Among other things, we prove theTheorem (Theorem II, below). Assume v is also volume preserving and that _x^'(t) 0 for all x M and real t; then, if _x^t: M M is weakly missng for some t, it is necessary that vx 0 at all x M.