This thesis calculates the Scalar curvature by expanding Christoffel symbols, so we get the relation about Scalar curvatures under conformal metrics. Then, we classify two-dimension Riemannian manifolds by Euler number and discuss the existence of the conformal metrics in the different Euler numbers. Finally, in the case of χ(M ) > 0, we give more details about the Trüdinger constant and see the possibilities for the different Trüdinger constants.

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