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We give a necessary condition of geometric nature for the existence of the$H$-fractional Brownian field indexed by a Riemannian manifold. In the case ofthe L\'evy Brownian field ($H=1/2$) indexed by manifolds with minimal closedgeodesics it turns out to be very strong. In particular we show that compactmanifolds admitting a L\'evy Brownian field are simply connected. We alsoderive from our result the nondegenerescence of the L\'evy Brownian fieldindexed by hyperbolic spaces. These results stress the need for alternativekernels on nonsimply connected manifolds to allow for Gaussian modelling orkernel machine learning of functional data with manifold-valued entries.

On the existence of fractional Brownian fields indexed by manifolds with closed geodesics