Obstruction to quasi-invariance of Gaussian measures under transport flows on Riemannian Manifolds

We prove an obstruction to quasi-invariance of Gaussian fields under divergence free transport flows on Riemannian manifolds. For the centered Gaussian field with covariance $(1-Δ_g)^{-α}$, $α>\frac{d}{2}$, quasi-invariance under the transport flow is equivalent to invariance. This happens precisely when the flow acts by isometries, or equivalently when the underlying vector field is Killing. The proof leverages the Feldman--Hájek theorem and utilizes localized pseudo-differential computations. On $\mathbb R^d$ this leaves only rigid motions, while on flat tori this leaves only translations.