  We show in this paper that there exists no $H$-fractional Brownian field
indexed by the cylinder $\mathbb{S}^1 \times ]0,\varepsilon[$ endowed with its
product distance $d$ for any $\varepsilon>0$ and $H>0$. This is equivalent to
say that $d^{2H}$ is not a negative definite kernel, which also leaves us
without a proof that many classical stationary kernels, such that the Gaussian
and exponential kernels, are positive definite kernels -- or covariances -- on
the cylinder.
  We generalise this result from the cylinder to any Riemannian Cartesian
product with a minimal closed geodesic. We also investigate the case of the
cylinder endowed with a distance asymptotically close to the product distance
in the neighbourhood of a circle.
  Another consequence is the discontinuity of the set of $H$ such that $d^{2H}$
is negative definite with respect to the Gromov-Hausdorff convergence on
compact metric spaces.
  These results extend our comprehension of kernel construction on metric
spaces, and in particular call for alternatives to classical kernels to allow
for Gaussian modelling and kernel method learning on cylinders.
