陳瑞堂Chen, Jui-Tang戴伯儒Dai, Bo-Ru2023-12-082022-06-212023-12-082022https://etds.lib.ntnu.edu.tw/thesis/detail/fa76805f69b42ec33e5c934858dbc705/http://rportal.lib.ntnu.edu.tw/handle/20.500.12235/121088noneThis 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.noneChristoffel symbolsScalar curvaturesRiemannian manifoldTrüdinger constantetd