  We consider the Poisson Boolean model of continuum percolation on a
homogeneous Riemannian manifold $M$. Let $lambda$ be intensity of the Poisson
process in the model and let $lambda_u$ be the infimum of the set of
intensities that a.s. produce a unique unbounded component. We show that above
$\lambda_u$ there is a.s. a unique unbounded component. We also study what
happens at $\lambda_u$ for some spaces. In particular, if $M$ is the product of
the hyperbolic disc and the real line, then at $\lambda_u$ there is a.s. not a
unique unbounded component. The results are inspired by results for Bernoulli
bond percolation on graphs due to Haggstrom, Peres and Schonmann.
