= Quadratic-form criterion for an Anosov flow
{title2=$\mathcal L_F Q>0\text{ on }TN/\mathbb RF$}
Let $F$ generate a nonsingular <smooth flow> on a <closed manifold>. Suppose a continuous nondegenerate indefinite <quadratic form> on the transverse <vector bundle> $TN/\mathbb RF$ has constant signature, is differentiable along the linearized flow, and has continuous positive-definite derivative along that flow. Then the flow is an <Anosov flow>. Compactness makes the positivity uniform; the positive and negative cones give the unstable and stable transverse directions. The flow direction must be quotiented out: its derivative cannot satisfy strict positivity.
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