  We show that there exists a duality between the local coordinates and the
solutions of the Klein-Gordon equation in curved spacetime in the same sense as
in the Minkowski spacetime. However, the duality in curved spacetime does not
have the same generality as in flat spacetime and it holds only if the system
satisfies certain constraints. We derive these constraints and the basic
equations of duality and discuss the implications in the quantum theory.

General Relativity and Quantum Cosmology;High Energy Physics - Theory