  Using an imbedding supported background tensor approach for the differential
geometry of an imbedded surface in an arbitrary background, we show that the
topological terms associated with the inner and outer curvature scalars of the
string worldsheet, have a dramatic effect on the global symplectic geometry of
the phase space of the theory. By identifying the global symplectic potential
of each Lagrangian term in the string action as the argument of the
corresponding pure divergence term in a variational principle, we show that
those topological terms contribute explicitly to the symplectic potential of
any action describing strings, without modifying the string dynamics and the
phase space itself. The variation (the exterior derivative on the phase space)
of the symplectic potential generates the integral kernel of a covariant and
gauge invariant symplectic structure for the theory, changing thus the global
symplectic geometry of the phase space. Similar results for non-Abelian gauge
theories and General Relativity are briefly discussed
