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Inspired by recent studies on string theory with non-geometric fluxes, wedevelop a differential geometry calculus combining usual diffeomorphisms withwhat we call beta-diffeomorphisms. This allows us to construct a manifestlybi-invariant Einstein-Hilbert type action for the graviton, the dilaton and adynamical (quasi-)symplectic structure. The equations of motion of thissymplectic gravity theory, further generalizations and the relation to theusual form of the string effective action are discussed. The Seiberg-Wittenlimit, known for open strings to relate commutative with non-commutativetheories, makes an interesting appearance.

A bi-invariant Einstein-Hilbert action for the non-geometric string