  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 of
the L\'evy Brownian field ($H=1/2$) indexed by manifolds with minimal closed
geodesics it turns out to be very strong. In particular we show that compact
manifolds admitting a L\'evy Brownian field are simply connected. We also
derive from our result the nondegenerescence of the L\'evy Brownian field
indexed by hyperbolic spaces. These results stress the need for alternative
kernels on nonsimply connected manifolds to allow for Gaussian modelling or
kernel machine learning of functional data with manifold-valued entries.
