= Solution
Here is the <quadratic-form criterion for an Anosov flow>. Assume that the generator $F$ is nowhere zero and form the transverse <vector bundle>
$$
\mathcal E=TN/\mathbb RF.
$$
Since $D\phi_tF(x)=F(\phi_t x)$, the derivative induces a linear cocycle $P_t:\mathcal E_x\to\mathcal E_{\phi_t x}$. Suppose there is a continuous nondegenerate indefinite <quadratic form> $Q_x$ on each fibre, with constant <signature of a quadratic form>, such that $t\mapsto Q_{\phi_t x}(P_t\xi)$ is differentiable and
$$
\mathcal DQ_x(\xi)=\left.\frac{d}{dt}\right|_{t=0}Q_{\phi_t x}(P_t\xi)
$$
is continuous and \b[positive definite on every transverse fibre]. Then $\phi_t$ is an <Anosov flow>. An equivalent way to specify $Q$ is a <quadratic form> on $TN$ whose radical is exactly $\mathbb RF$ and whose induced form and derivative satisfy these conditions on the quotient.
On the <closed manifold>, compactness makes the positivity uniform: for some $c>0$, $\mathcal DQ_x(\xi)\ge c\|\xi\|^2$. Thus the positive and negative cones are strictly separated by the linearized flow, yielding uniform unstable and stable transverse subbundles and their invariant lifts. The positive index is the unstable dimension and the negative index is the stable dimension. Strict positivity must be imposed transversely; imposing it on the flow direction would be impossible because that direction is carried into itself.
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