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We formulate and solve the analog of the universal Conformal Ward Identityfor the stress-energy tensor on a compact Riemann surface of genus $g>1$, andpresent a rigorous invariant formulation of the chiral sector in the inducedtwo-dimensional gravity on higher genus Riemann surfaces. Our construction ofthe action functional uses various double complexes naturally associated with aRiemann surface, with computations that are quite similar to descentcalculations in BRST cohomology theory. We also provide an interpretation forthe action functional in terms of the geometry of different fiber spaces overthe Teichm\"{u}ller space of compact Riemann surfaces of genus $g>1$.Comment: 38 pages. Latex2e + AmsLatex2.1. One embedded figure. One section on  the relation with the geometry of fiber spaces on the Teichmueller space and  several important references adde

Generating Functional in CFT and Effective Action for Two-Dimensional Quantum Gravity on Higher Genus Riemann Surfaces