  A set of on shell duality equations is proposed that leads to a map between
strings moving on symmetric spaces with opposite curvatures. The transformation
maps "waves" on a riemannian symmetric space to "waves" on its dual riemannian
symmetric space. This transformation preserves the energy momentum tensor
though it is not a canonical transformation. The preservation of the energy
momentum tensor has a natural geometrical interpretation. The transformation
maps "particle-like solutions" into static "soliton-like solutions". The
results presented here generalize earlier results of E. Ivanov.
