= Pestov identity for a geodesic flow
{c}
{title2=$2\langle H\varphi,VX\varphi\rangle=\|X\varphi\|^2+\|H\varphi\|^2-\int K(V\varphi)^2$}
On a closed oriented Riemannian surface this integral identity follows from the canonical commutators and integration by parts against <Liouville volume of a surface geodesic flow>. If $VX\varphi=-2H\varphi$ and $K<0$, substitution gives $\|X\varphi\|^2+5\|H\varphi\|^2+\int(-K)(V\varphi)^2=0$. Each term is nonnegative, forcing all three derivatives to vanish. For $\varphi=V^2u+u$, the fibre equation then makes $u$ a constant plus a fibrewise linear function, supplying the <potential symmetric 2-tensor> kernel.
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