Geometricians before Gauss always associate the surface with the peripheral space E3 when studying the surface, find out the principal direction of a point on the surface, and then calculate the product of the normal curvature of two curvature lines, which is Euler's research. Gauss proved that the total curvature of a surface is determined by the first basic form of the surface, which is the Gauss equation, so the total curvature is usually called Gauss curvature, which is a famous discovery of Gauss and called "fantastic theorem". He said: "If a surface can be extended to any other surface, then the curvature of each point remains the same." The "developable" here means mapping to 1- 1 (one-to-one correspondence) and keeping a distance. The intrinsic geometry established by Gauss has far-reaching influence, which is a key and significant breakthrough in differential geometry, but it was not recognized by people at that time. The more important development belongs to the German mathematician (G.F.) B. Riemann. 1854, he delivered an inaugural speech entitled "On Hypothesis as the Basis of Geometry" at the University of G? ttingen. Riemann regards the surface itself as an independent geometric entity, not just a geometric entity in Euclidean space. He developed the concept of space. First of all, he put forward the concept of N-dimensional manifold (called multiple extensions at that time), in which the points are n real numbers (x 1, x2, ..., xn), and he defined two infinitely adjacent points (xi+dxi) (i = 1, 2, ...) on the manifold. Later, (2) is called Riemann metric, where (gij) is a positive definite symmetric matrix. Riemann recognizes that metric (2) is a structure added to a manifold, so there can be many Riemann metrics on the same manifold. Geometricians before Riemann only knew that the measure of the peripheral space E3 gave the surface S an induced measure, and (3) was the first basic form, but they didn't realize that the surface S could be defined independently of E3 and given a metric structure. Riemann realized that this was extremely important, and he separated the induced metric from the independent Riemann metric, thus creating Riemann geometry with (2) as the starting point. This geometry takes various non-Euclidean geometries as its special cases. For example, (α is a constant) (4) can be taken as the distance between two infinite adjacent points. When α >: 0 is spherical geometry or elliptic geometry (also known as the geometry of normal curvature space), α=0 is Euclidean geometry.
A basic problem in Riemannian geometry is the equivalence of differential forms. In two different coordinate systems x 1, x2, ..., xn and x 1', x2', ..., xn', given the sum of two quadratic differential forms, the condition that there is coordinate change (I = 1, 2, ... n) will change one differential form into the other. This question is 60. Christophe solution includes the notation named after him, namely the first kind of Christophe notation jk, L and the second kind of Christophe notation:, (5) and the concept of covariant differential (see Riemann geometry). On this basis, from 1887 to 1896, G. Rich developed tensor analysis method, which played a fundamental role in general relativity. Ritchie and his student T. Levi-Zivita made a detailed summary of Ritchie's calculation method in their research report Absolute Differential Method and Its Application (190 1).