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An Anosov flow is a nonsingular smooth flow on a compact smooth manifold with a continuous derivative-invariant splittingHere $F$ is the generating vector field, the flow direction is one-dimensional, and in the usual nontrivial convention both $E^s$ and $E^u$ have positive dimension. Constants $C,\lambda >0$ must work uniformly over all points, all $t\ge 0$ and the corresponding tangent vectors:The stable bundle of an Anosov flow contracts forward; the unstable bundle of an Anosov flow contracts backward. The neutral direction along $F$ is essential: changing the starting point slightly along an orbit mainly shifts its clock, so that direction cannot be included in either uniform contraction estimate.

A suspension flow of the cat map gives a three-dimensional example. A positive time change of a smooth flow preserves the Anosov property on a compact manifold, although its stable and unstable bundles can tilt along the flow direction. These are uniform structural conditions, stronger than a sign observed in one finite-time Lyapunov exponent calculation.

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.

For relativistic dust, $p=0$, the relativistic Euler equation gives $\rho a_b=0$. The nowhere-vanishing density therefore makes the four-acceleration zero: $u$ is a timelike geodesic vector field. Let $\alpha =u^{♭}$ be the differential one-form obtained by the musical isomorphism. Its exterior derivative satisfiesHere ${\iota }_u$ is the interior product. By hypersurface orthogonality and the Frobenius theorem, $\alpha \wedge d\alpha =0$. Taking the interior product and using $\alpha (u)=-1$ givesThus $\boxed{d{u}^{♭}=0}$. By the Poincaré lemma, locally $\alpha =-dt$, and then $u(t)=-\alpha (u)=1$. Choose spatial coordinate functions $x^i$ on one level set of $t$ and extend them constantly along the integral curves of a vector field $u$, using its smooth flow. This constructs $u={\partial }_t$. Since $dt({\partial }_i)=0$,The metric on the orthogonal complement of a timelike vector is positive-definite, so the synchronous coordinates for irrotational dust haveThis construction is local; crossing geodesics can prevent these synchronous coordinates from covering a larger region.

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 a nonsingular smooth flow on a closed manifold $N$ be generated by $F$. Suppose a continuous nondegenerate indefinite quadratic form $Q$ on the transverse vector bundle $TN/\mathbb{R}F$ has constant signature, is differentiable along the linearized flow, and has continuous positive-definite derivative there. Then the flow is an Anosov flow.^[ref]

Here differentiating the form means differentiating $Q_{{\Phi }_{t}x}(D{\Phi }_{t}v)$. Compactness gives constants $M,c>0$ such that $|Q(v)|\le M\Vert v{\Vert }^2$ and $\dot{Q}(v)\ge c\Vert v{\Vert }^2$. On the positive cone,so the form increases at least exponentially. Negative cones supply the analogous backward-time estimate. Strict cone invariance produces uniformly expanding and contracting transverse bundles; the full splitting also contains the neutral flow direction.

For a geodesic flow on a compact negatively curved surface, a transverse Jacobi field has scalar coordinates satisfying $\dot{y}=z$ and $\dot{z}=-K(t)y$, with Gaussian curvature $K\le -\kappa <0$. The indefinite form $Q(y,z)=yz$ obeysfor every nonzero transverse vector. This directly verifies the criterion. The quotient by $F$ is essential: $D{\Phi }_{t}F=F\circ {\Phi }_t$ gives the neutral direction that cannot satisfy this strict transverse positivity. The estimates concern the form; constructing the invariant bundles is the additional cone argument, rather than an assertion that every positive-cone vector has monotonically growing norm.

A smooth flow on a smooth manifold is a smooth map $(t,x)\mapsto {\Phi }_t(x)$ withThe identity implies ${\Phi }_{-t}={\Phi }_t^{-1}$, so each time map is a diffeomorphism. Its generating vector field is $F(x)={\partial }_t{\Phi }_t(x){|}_{t=0}$; trajectories solve the autonomous differential equation $\dot{x}=F(x)$.

For example, ${\Phi }_t(x)=x+t$ is a smooth flow on the real line, while ${\Phi }_t(\theta )=\theta +t(\mathrm{mod}2\pi )$ is a rotation flow on a circle. The second has periodic trajectories, even though the same additive time parameter acts in both examples.

A smooth vector field initially gives only a local flow. The equation $\dot{x}=x^2$ has $x(t)=x_0/(1-x_{0}t)$, which blows up at $t=1/x_0$ for $x_0>0$. On a closed manifold, compactness prevents escape in finite time and local existence can be continued for every real time. Thus every smooth vector field there generates a global smooth flow. On a noncompact manifold, global existence is a separate property.

For relativistic dust with nowhere-zero density, stress-energy conservation implies zero four-acceleration. If its normalized four-velocity $u$ has hypersurface orthogonality, then $u^{♭}\wedge d{u}^{♭}=0$ and ${\iota }_{u}d{u}^{♭}=0$. The interior product of the first identity gives $d{u}^{♭}=0$. The Poincaré lemma supplies $u^{♭}=-dt$ locally; spatial coordinate functions constant along the smooth flow of $u$ give $u={\partial }_t$, $g_{tt}=-1$ and $g_{ti}=0$. The resulting synchronous coordinates describe the dust until the local coordinate construction fails.