  The classical Smorodinsky-Winternitz systems on the ND sphere, Euclidean and
hyperbolic spaces S^N, E^N and H^N are simultaneously approached starting from
the Lie algebras so_k(N+1), which include a parametric dependence on the
curvature k. General expressions for the Hamiltonian and its integrals of
motion are given in terms of intrinsic geodesic coordinate systems. Each Lie
algebra generator gives rise to an integral of motion, so that a set of
N(N+1)/2 integrals is obtained. Furthermore, 2N-1 functionally independent ones
are identified which, in turn, shows that the well known maximal
superintegrability of the Smorodinsky-Winternitz system on E^N is preserved
when curvature arises. On both S^N and H^N, the resulting system can be
interpreted as a superposition of an "actual" oscillator and N "ideal"
oscillators (for the sphere, these are alike the actual ones), which can also
be understood as N "centrifugal terms"; this is the form seen in the Euclidean
limiting case.
