= Solution
Let $F$ be the generator of the <smooth flow>. The <Anosov flow> property requires $F$ to be nowhere zero and a continuous splitting of the <tangent bundle>
$$
TN=E^s\oplus\mathbb RF\oplus E^u
$$
which is invariant under $D\phi_t$. There must be constants $C\ge1$ and $\kappa>0$ such that, for every $x$, every $t\ge0$ and every vector in the indicated fibre,
$$
\boxed{\|D\phi_t\xi_s\|\le Ce^{-\kappa t}\|\xi_s\|,
\qquad
\|D\phi_{-t}\xi_u\|\le Ce^{-\kappa t}\|\xi_u\|.}
$$
The first inequality defines the <Anosov stable bundle>; the second defines the <Anosov unstable bundle> and is equivalent to uniform forward expansion there. The remaining one-dimensional direction is the flow direction. The <norm> can come from any <Riemannian metric>, since norms are uniformly equivalent on a <closed manifold>. The essential requirements are a continuous invariant splitting and estimates uniform in both the base point and time, rather than separate exponential rates along individual orbits.
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