  We consider isotropic L\'evy processes on a compact Riemannian manifold,
obtained from an $\mathbb{R}^d$-valued L\'evy process through rolling without
slipping. We prove that the Feller semigroups associated with these processes
extend to strongly continuous contraction semigroups on $L^p$, for $1\leq
p<\infty$, and that they are self-adjoint when $p=2$. When the motion has a
non-trivial Brownian part, we prove that the generator has a discrete spectrum
of eigenvalues and that the semigroup is trace-class.
