Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 17 / 1 / b / Solution

Here is the quadratic-form criterion for an Anosov flow. Assume that the generator $F$ is nowhere zero and form the transverse vector bundleSince $D{\varphi }_{t}F(x)=F({\varphi }_{t}x)$, the derivative induces a linear cocycle $P_t:{\mathcal{E}}_x\to {\mathcal{E}}_{{\varphi }_{t}x}$. Suppose there is a continuous nondegenerate indefinite quadratic form $Q_x$ on each fibre, with constant signature of a quadratic form, such that $t\mapsto Q_{{\varphi }_{t}x}(P_t\xi )$ is differentiable andis continuous and positive definite on every transverse fibre. Then ${\varphi }_t$ is an Anosov flow. An equivalent way to specify $Q$ is a quadratic form on $TN$ whose radical is exactly $\mathbb{R}F$ and whose induced form and derivative satisfy these conditions on the quotient.

On the closed manifold, compactness makes the positivity uniform: for some $c>0$, $\mathcal{D}{Q}_x(\xi )\ge c\Vert \xi {\Vert }^2$. 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.