根據高斯-博內定理，我們可以導出符號條件。這裡有一個很自然的問題：符號條件的逆敘述是否正確？在這篇論文中，我們將導出平均場方程，並且討論當歐拉示性數為0時，在黎曼曲面上的曲率問題。

According the Gauss-Bonnet theorem, we can derive the sign conditions. There is a natural question: is the converse of the sign conditions true? In this thesis, we shall derive the mean field equation and discuss the curvaure problems on a Riemannian surfaces when the Euler characteristic is 0.

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