= Solution
For a nonsingular <smooth flow> generated by $F$ on a <closed manifold>, the <Anosov flow> condition is a continuous $D\phi_t$-invariant splitting
$$
TN=E^s\oplus\mathbb RF\oplus E^u
$$
and constants $C,\lambda>0$ such that, for all $t\ge0$,
$$
\boxed{\|D\phi_t v_s\|\le Ce^{-\lambda t}\|v_s\|,\qquad
\|D\phi_{-t}v_u\|\le Ce^{-\lambda t}\|v_u\|}.
$$
The <norm> can come from any auxiliary <Riemannian metric>; compactness makes the resulting definitions equivalent. The centre direction is the flow direction, with $F$ nowhere zero.
One construction is a <suspension flow>. The <hyperbolic toral automorphism> of $\mathbb T^2$ defined by
$$
A=\begin{pmatrix}2&1\\1&1\end{pmatrix}
$$
has <eigenvalues> $\lambda_\pm=(3\pm\sqrt5)/2$, with $0<\lambda_-<1<\lambda_+$. Form the closed three-dimensional <mapping torus> $(\mathbb T^2\times[0,1])/((x,1)\sim(Ax,0))$ and move in the second coordinate at unit speed. Each crossing of the section applies $A$. The two one-dimensional <eigenspaces> give invariant transverse bundles, and after $k$ crossings their derivatives multiply by $\lambda_-^k$ and $\lambda_+^k$. The difference between $k$ and elapsed time is bounded, so any smooth <Riemannian metric> gives the required uniform exponential estimates. The vertical direction supplies the centre line.
A different construction is the <geodesic flow> on the <unit tangent bundle> of a closed oriented <hyperbolic surface>. This bundle is a closed three-manifold. In curvature $-1$, a normal <Jacobi field> along a unit-speed geodesic satisfies $j''-j=0$. The initial data $(j,j')=(a,-a)$ give $j(t)=ae^{-t}$, while $(a,a)$ give $j(t)=ae^t$. The linearized <geodesic flow> transports these initial data as $(j(t),j'(t))$, so they give one stable and one unstable direction with precisely those exponential rates. The remaining tangent direction is the geodesic <vector field>. Thus this also constructs an <Anosov flow>, using geometric rather than return-map hyperbolicity.
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