  We derive an explicit formula for the Jacobi field that is acting in an
extended Fock space and corresponds to an ($\R$-valued) L\'evy process on a
Riemannian manifold. The support of the measure of jumps in the
L\'evy--Khintchine representation for the L\'evy process is supposed to have an
infinite number of points. We characterize the gamma, Pascal, and Meixner
processes as the only L\'evy processes whose Jacobi field leaves the set of
finite continuous elements of the extended Fock space invariant.
