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An Anosov diffeomorphism is a continuously differentiable diffeomorphism of a compact smooth manifold with an invariant splitting of its tangent bundle into stable and unstable vector subbundles satisfying the displayed uniform exponential estimates for $n\ge 0$, with $C,\lambda >0$. Hyperbolic toral automorphisms give examples. An Anosov flow has a further one-dimensional invariant bundle along the flow direction; its definition is a distinct continuous-time version of uniform hyperbolicity.

A smooth function $h$ is a flow coboundary if $h=Fu$ for some smooth function $u$, where $F$ is the flow generator. Equivalently $u({\varphi }_{t}x)-u(x)={\int }_0^{t}h({\varphi }_{s}x)ds$. Thus its integral over every periodic orbit is zero. For a transitive Anosov flow, the Livsic theorem supplies the converse and smooth regularity.

For a smooth Anosov flow generated by a surface thermostat on a closed oriented Riemannian surface, a smooth base function plus a smooth one-form is a smooth flow coboundary precisely when the base function is zero and the one-form is an exact differential form. This is the nontrivial degree-zero/one kernel theorem. It does not require negative curvature or preservation of Liouville volume. The converse follows directly from $F(f\circ \pi )=df(v)$. A primary proof of the kernel theorem is arxiv.org/abs/math/0601039.

For a topologically transitive Anosov flow, a Hölder function has a Hölder flow coboundary precisely when its integral around every periodic orbit vanishes. The coboundary identity is $u({\varphi }_{t}x)-u(x)={\int }_0^{t}h({\varphi }_{s}x)ds$. For smooth flow and smooth $h$, the smooth regularity theorem gives smooth $u$. Necessity follows by setting $t$ equal to a period. The solution is unique up to a constant by density of an orbit.

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.

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.

The Livsic theorem says that for a topologically transitive Anosov flow on a closed manifold, a Hölder continuous function $f$ is a flow coboundary if and only if it has zero integral over every periodic orbit. In the integral formulation there is a Hölder continuous function $u$ satisfyingEquivalently, $u$ is differentiable along the flow and $Fu=f$. The periodic obstruction isIt is necessary because the left-hand side of the coboundary identity is zero at a period. The substantive theorem is its sufficiency. For a smooth flow and smooth $f$, the smooth regularity version gives a smooth $u$. Two such solutions differ by a constant: their difference is flow invariant and hence constant on a dense orbit and, by continuity, everywhere. This smooth version is the one needed in part (b).

Evaluate the smooth one-form on unit velocities to obtain $f(x,v)={\theta }_x(v)$. Its integral along a periodic orbit of the geodesic flow is exactly the one-form integral along the corresponding closed geodesic. Negative curvature makes this geodesic flow an Anosov flow; the geodesic flow of a connected closed negatively curved surface is also topologically transitive. Apply the smooth Livsic theorem to obtain $u\in C^{\infty }(SM)$ withIf there is more than one connected component, apply this argument to each component separately.

The vertical vector field of a surface unit tangent bundle differentiates rotation of $v$, soFor each $x$, writing ${\theta }_x(v)=A\cos \vartheta +B\sin \vartheta$ shows explicitly thatThe fibre disintegration of Liouville volume of a surface geodesic flow is $d\mu =d{A}_{g}d\vartheta$. Integrating the displayed equality over the base therefore gives $\Vert VXu{\Vert }_{L^2}^2=\Vert Xu{\Vert }_{L^2}^2$. Substitution into the supplied Pestov identity for a geodesic flow yieldsBoth terms are nonnegative and $-K\ge k_0>0$. Thus $Vu=0$ everywhere. Since every unit-circle fibre is connected, $u$ is independent of velocity and descends to a smooth function $h$ on $M$: $u=h\circ \pi$. Its geodesic derivative isComparison with $Xu={\theta }_x(v)$, first on unit vectors and then on all vectors by linearity, provesThus the one-form is an exact differential form. This proves the degree-one geodesic X-ray transform conclusion directly from the given integral identity and the stated smooth coboundary theorem.

Use the following permitted kernel result, stated in the form needed here. For an Anosov flow generated by a smooth surface thermostat on a closed oriented Riemannian surface, if a smooth function $u$ on $SM$ satisfieswhere $h$ is a smooth base function and $\eta$ is a smooth one-form, then $h=0$ and $\eta$ is an exact differential form. Conversely these two conditions give a smooth solution by taking a base primitive. This is the kernel of the thermostat X-ray transform in degrees zero and one. In particular it applies to Gaussian thermostats without an additional curvature condition; it is stronger than the magnetic or geodesic special case.

Put $\theta =E^{♭}$, so ${\theta }_x(v)=⟨E(x),v⟩$. Suppose the thermostat preserves a smooth volume form. Write it as $\rho \mu$ with smooth $\rho >0$, choosing its sign separately on each component if necessary. By the invariant volume criterion for a smooth flow and the divergence computation in part (b),Thus $u=\log \rho$ is a smooth solution of $Fu={\theta }_x(v)$. Apply the stated kernel theorem with $h=0$ to obtain $\theta =df$ for a smooth base function $f$. The definition of the gradient then gives $E={\mathrm{grad}}_{g}f$.

Conversely, suppose $E={\mathrm{grad}}_{g}f$. Since $d\pi (F)=v$, the vertical part has no effect on a base function andThe divergence of a Gaussian thermostat and the volume-change formula therefore giveConsequentlyOnly the necessary direction uses the Anosov kernel theorem; the displayed invariant density works for any potential Gaussian thermostat. In particular its exponent is positive, as required to cancel the negative divergence found in part (b).

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.

The Livsic theorem for a topologically transitive Anosov flow says that a Hölder function $h:N\to \mathbb{R}$ satisfies the periodic-orbit obstructionif and only if there is a Hölder function $u$ satisfying the flow coboundary identityThe solution is differentiable along orbits and $Fu=h$ in that sense. For a smooth flow and smooth $h$, the smooth regularity version gives $u\in C^{\infty }(N)$, so $Fu=h$ holds smoothly. The solution is unique up to an additive constant: the difference of two solutions is invariant and must be constant on a dense orbit and then everywhere by continuity. Necessity of the periodic obstruction follows immediately by taking $t=T$ in the displayed identity.

The geodesic flow of a closed negatively curved surface is a topologically transitive Anosov flow. In particular, its Jacobi-field hyperbolicity and the usual negative-curvature transitivity theorem allow the smooth Livsic theorem to be applied. Set $b(x,v)={\beta }_x(v,v)$. The given closed-geodesic integrals are exactly its periodic-orbit obstructions on $SM$. Therefore

We make the differential operators and signs explicit. On the unit tangent bundle, let $V$ generate positive fibre rotation, $X$ be the geodesic vector field and $H=[V,X]$. For the canonical coframe of a surface unit tangent bundle, writeand let $\omega$ be the Levi-Civita connection form with $\omega (V)=1$. This uses $\theta$ for a one-form, keeping $\beta$ for the given tensor. The dual frame is $(X,H,V)$, and the structure equations areThese follow by writing the two horizontal forms as a rotated local orthonormal coframe and $\omega =d\vartheta +{\omega }_0$, with ${\omega }_0$ its connection 1-form. Equivalently, differentiating the sine and cosine components of $X$ twice in the fibre angle givesThe curvature equation also gives $[X,H]=KV$. The Liouville volume of a surface geodesic flow will be taken positive in the base/positive-fibre orientation, namely $\mu =\alpha \wedge \theta \wedge \omega$.

In an oriented orthonormal frame at $x$, put $v=(\cos \vartheta ,\sin \vartheta )$. The symmetric tensor evaluation isSince $V={\partial }_{\vartheta }$, this proves $(V^3+4V)b=0$.

The Lie bracket of vector fields identities give an operator identity that supplies the required equation. First,Using $VH=HV-X$ and $XV=VX-H$, differentiate this expression once more to obtainFor $\phi =V^{2}u+u$ and $Xu=b$, the first term vanishes by the explicit fibre calculation. Hence

Write $\rho =1/f$ and $t={\int }_0^s\rho ({\varphi }_{r}x)dr$. For an Anosov stable bundle, the new stable directions are $v+{\lambda }_s(x,v)F(x)$, where ${\lambda }_s=\rho (x{)}^{-1}{\int }_0^{\infty }d{\rho }_{{\varphi }_{r}x}(D{\varphi }_{r}v)dr$. Exponential contraction makes this integral uniformly convergent. Differentiating the clock equation shows that its flow-direction coefficient after time $t$ is $\rho ({\varphi }_{s}x{)}^{-1}{\int }_s^{\infty }d\rho (D{\varphi }_{r}v)dr$, which decays exponentially and has the same graph form at the new point. Negative-time integrals give the unstable graph. Compactness and positivity of $f$ compare old and new times uniformly, preserving the Anosov flow property.

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.

A suspension flow moves vertically in the mapping torus of a diffeomorphism, applying that diffeomorphism whenever an orbit crosses the identifying section. A hyperbolic toral automorphism with constant roof gives a three-dimensional Anosov flow: after $k$ crossings its transverse derivative is $A^k$, while the remaining fractional time contributes a uniformly bounded factor.

The unstable bundle consists of transverse directions exponentially contracted in negative time. Together with the stable bundle and the flow direction it gives the invariant splitting of an Anosov flow. Under a positive time change its graph coefficient is obtained from the convergent past integral $-\rho (x{)}^{-1}{\int }_{-\infty }^{0}d\rho (D{\varphi }_{r}v)dr$.