This theorem is an important result in differential geometry, which describes the relationship between geometric properties of surfaces and their curvatures.
Gauss-Bonne theorem is expressed as follows:
For compact surface m, the difference between the integral of Gaussian curvature k on the whole surface and the sum of average curvature on the surface boundary is equal to 2π. Namely:
∫∫M K dA + ∫∫？ M k_g ds = 2π，
Where ∫∫M represents the surface integral on the surface m, and k is the Gaussian curvature of the surface. M represents the boundary of surface m, ∫∫? M represents the surface integral on the surface boundary, k_g represents the average curvature on the surface boundary, ds represents the line element on the boundary, and 2π is a constant.
This theorem shows the relationship between the geometric properties of a surface (through Gaussian curvature) and its topological properties (through the surface boundary). Through this theorem, we can judge the topological type and the overall geometric properties of the surface by calculating the Gaussian curvature and the average curvature on the surface boundary.
Gauss-Bonnet theorem has important application and significance in the study of differential geometry and surface theory, and is widely used in the classification of surfaces, geodesic theory of surfaces and other fields.
The history of the basic theorem of differential geometric surface theory
The basic theorem of differential geometry is an important result about the curvature and topological properties of surfaces. German mathematician Karl Gauss first proposed and proved it in 1827.
At the end of 18 and the beginning of 19, European mathematicians began to study the properties and curvature of surfaces. French mathematician Pierre Louis Lagrange and Jean-Joseph the Baptist made great contributions in this field. But their research mainly focuses on two-dimensional plane curves, not three-dimensional surfaces.
During this period, the study of Gauss had a great influence on the development of differential geometry. He was one of the first mathematicians to study three-dimensional surfaces, and he proposed a comprehensive method to study the curvature and topological properties of surfaces. Gauss put forward the basic theorem of surfaces for the first time in his paper "General Research on Surfaces" published in 1827.
The basic theorem of gauss points out that the total curvature of any surface is equal to the product of gauss curvature and average curvature on the surface. This theorem shows the relationship between curvature and geometric properties on a surface, which is of great significance for understanding the shape and properties of a surface.
Gauss's research had a far-reaching influence on the development of differential geometry, and provided important methods and ideas for later mathematicians. His basic theorem not only plays an important role in the study of differential geometry, but also has been widely used in physics and engineering.