 Quadratic-form criterion for an Anosov flow

Let a nonsingular smooth flow on a closed manifold $N$ be generated by $F$. Suppose a continuous nondegenerate indefinite quadratic form $Q$ on the transverse vector bundle $TN/\mathbb{R}F$ has constant signature, is differentiable along the linearized flow, and has continuous positive-definite derivative there. Then the flow is an Anosov flow.^[ref]

Here differentiating the form means differentiating $Q_{{\Phi }_{t}x}(D{\Phi }_{t}v)$. Compactness gives constants $M,c>0$ such that $|Q(v)|\le M\Vert v{\Vert }^2$ and $\dot{Q}(v)\ge c\Vert v{\Vert }^2$. On the positive cone,so the form increases at least exponentially. Negative cones supply the analogous backward-time estimate. Strict cone invariance produces uniformly expanding and contracting transverse bundles; the full splitting also contains the neutral flow direction.

For a geodesic flow on a compact negatively curved surface, a transverse Jacobi field has scalar coordinates satisfying $\dot{y}=z$ and $\dot{z}=-K(t)y$, with Gaussian curvature $K\le -\kappa <0$. The indefinite form $Q(y,z)=yz$ obeysfor every nonzero transverse vector. This directly verifies the criterion. The quotient by $F$ is essential: $D{\Phi }_{t}F=F\circ {\Phi }_t$ gives the neutral direction that cannot satisfy this strict transverse positivity. The estimates concern the form; constructing the invariant bundles is the additional cone argument, rather than an assertion that every positive-cone vector has monotonically growing norm.