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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 magnetic flow on a closed oriented surface, a smooth function plus a smooth one-form can be a smooth flow coboundary only when the function is zero and the one-form is exact. Equivalently their sum has zero integral along every closed orbit precisely in that case, using the smooth Livsic theorem. This is the nontrivial magnetic transform kernel theorem. Exact one-forms give the converse directly, since $F(w\circ \pi )=dw(v)$. The theorem does not require pointwise negative Gaussian curvature.

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.

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