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The interplay between disorder and quantum interference leads to a widevariety of physical phenomena including celebrated Anderson localization -- thecomplete absence of diffusive transport due to quantum interference betweendifferent particle trajectories. In two dimensions, any amount of disorder isthought to induce localization of all states at long enough length scales,though this may be prevented if bands are topological or have strong spin-orbitcoupling. In this note, we present a simple argument providing anothermechanism for disrupting localization: by tuning the underlying curvature ofthe manifold on which diffusion takes place. We show that negative curvaturemanifolds contain a natural infrared cut off for the probability of selfreturning paths. We provide explicit calculations of the Cooperon -- directlyrelated to the weak-localization corrections to the conductivity -- inhyperbolic space. It is shown that constant negative curvature leads to a rapidgrowth in the number of available trajectories a particle can coherentlytraverse in a given time, reducing the importance of interference effects andrestoring classical diffusive behavior even in the absence of inelasticcollisions. We conclude by arguing that this result may be amenable toexperimental verification through the use of quantum simulators.

Absence of Weak Localization on Negative Curvature Surfaces