Infinitesimal obstructions to weakly mixing (Q1202524)

This paper considers the flow \(\varphi^ t\) of a smooth vector field \(v\) on a closed Riemannian manifold \(M^ n\) whose orbits are geodesics. The \((n-1)\)-plane field normal to \(v\), denoted \(\perp v\), is invariant under \(d\varphi^ t\). The authors define a smooth real function \(\Lambda_ x(t):\mathbb{R}\to\mathbb{R}\) for each \(x\in M^ n\) by \[ \Lambda_ x(t)=\log\prod^{n-1}_{i=1}(1+\lambda_ i(t)) \] where \(\lambda_ i(t)\) are the eigenvalues of \(AA^ T\) and \(A\) is the matrix (with respect to an orthonormal basis) of the nonsingular map \(d\varphi^{2t}\) restricted to \(\perp v\) at the point \(\varphi_ x^{-t}\in M^ n\). The author's main theorem is that if \(v\) is volume preserving, \(\Lambda_ x''(t)\geq 0\) for all \(x\in M^ n\) and all \(t\), and if \(\varphi_ x^ t:M^ n\to M^ n\) is weakly mixing for some \(t\), then \(\|\nabla v\|_ x\neq 0\) for all \(x\in M^ n\).