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 minimising 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 non-degenerescence of the L\'evy Brownian field indexed by hyperbolic spaces.
On the quasi-invariance of the Wiener measure on path spaces and the anticipative integrals over a Riemannian manifold
Bilaplacian on a Riemannian manifold and Levi-Civita connection
Derivative Formulas in Measure on Riemannian Manifolds
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