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A nonsingular smooth flow on a compact smooth manifold is Anosov if there is a continuous invariant splitting $TN=E^s\oplus \mathbb{R}F\oplus E^u$ and constants $C,\lambda >0$ with $\Vert D{\varphi }_t{v}_s\Vert \le C{e}^{-\lambda t}\Vert v_s\Vert$ and $\Vert D{\varphi }_{-t}{v}_u\Vert \le C{e}^{-\lambda t}\Vert v_u\Vert$ for $t\ge 0$. Uniform contraction and expansion are transverse to the neutral flow direction, which distinguishes the splitting from that of an Anosov diffeomorphism.

A smooth real function $\lambda$ on the unit tangent bundle determines this unit-speed smooth flow. Its base velocity is $v$, so its generator never vanishes. In the canonical frame its divergence in Liouville volume of a surface geodesic flow is $V\lambda$. The special case where $\lambda$ depends only on the base is a magnetic flow; when $\lambda (x,v)=⟨E(x),iv⟩$ it is a Gaussian thermostat.

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 bundlewhich is invariant under $D{\varphi }_t$. There must be constants $C\ge 1$ and $\kappa >0$ such that, for every $x$, every $t\ge 0$ and every vector in the indicated fibre,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.

Use the derivative to transport the planes:The derivative is an invertible linear map, so it sends a two-dimensional subspace to a two-dimensional subspace. The flow identity ${\varphi }_t\circ {\varphi }_s={\varphi }_{t+s}$ givesHence a plane containing the generator is sent to another plane containing the generator; the displayed formula is a well-defined smooth lift to $\Lambda (SM)$. The chain rule gives ${\varphi }_{t+s}^{\ast }={\varphi }_t^{\ast }\circ {\varphi }_s^{\ast }$, and ${\varphi }_0^{\ast }$ is the identity. Thus it is a genuine lifted smooth flow, with inverse ${\varphi }_{-t}^{\ast }$, rather than merely a family of maps of individual fibres. Its generator is $F^{\ast }={{\partial }_t{\varphi }_t^{\ast }|}_{t=0}$.

For a nonsingular smooth flow generated by $F$ on a closed manifold, the Anosov flow condition is a continuous $D{\varphi }_t$-invariant splittingand constants $C,\lambda >0$ such that, for all $t\ge 0$,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 byhas eigenvalues ${\lambda }_{\pm }=(3\pm \sqrt{5})/2$, 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 $j^{\prime \prime }-j=0$. The initial data $(j,j^{\prime })=(a,-a)$ give $j(t)=a{e}^{-t}$, while $(a,a)$ give $j(t)=a{e}^t$. The linearized geodesic flow transports these initial data as $(j(t),j^{\prime }(t))$, 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.

Let $F$ generate a nonsingular smooth flow on a closed manifold. Suppose a continuous nondegenerate indefinite quadratic form on the transverse vector bundle $TN/\mathbb{R}F$ 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.