我们首先回顾具有度量的二维微分流形（抽象曲面）的定义，然后在抽象曲面上回顾切向量场与协变导数的定义，从而引入其上的Christoffel 符号和Riemann 曲率张量，接着我们仿照Gauss 绝妙定理的形式，由Riemann 曲率张量和度量便可以得到抽象曲面上一个在不同局部坐标系下的不变的量，也就是Gauss 曲率。而以上提及的由度量决定的量均与三维空间中曲面上对应的由第一基本形式决定的内蕴量的形式相吻合。最后，我们来验证抽象曲面上Gauss 曲率与Euler 示性数的关系，即Gauss-Bonnet 定理。

      We first recall the definition of two-dimensional differential manifold with metric (i.e., abstract surfaces). Then on the abstract surface, we recall the definition of the tangent vector field, covariant derivative, Christoffel symbol and the Riemann curvature tensor. Then by imitating Gauss's perfect theorem, an important invariant quantity under different local coordinates named the Gauss curvature can be seen through the Riemann curvature tensor and metric. All the above quantities determined by the metric are consistent with the corresponding intrinsic quantities determined by the first fundamental form on a surface in three-dimensional Euclidean space. Finally, let's verify the relationship between Gauss curvature and Euler's characteristic number on the abstract surface, that is, the Gauss-Bonnet theorem.