A basic problem in Riemannian geometry is the equivalence of differential forms. This problem was solved by E.B. Christophel and R. Lipschitz around 1869. The solution of the former includes two kinds of Christophel symbols and the concept of covariant differential named after his surname. On this basis, G. Rich developed the tensor analysis method, which played the role of a basic mathematical tool in general relativity. They further developed Riemann geometry.

However, in Riemann's time, Lie groups and topology have not yet developed, so Riemann geometry is limited to a small range of theories. About 1925, H. hopf began to study the relationship between differential structure and topological structure of Riemannian space. With the establishment of the precise concept of differential manifold, especially in the 1920s, E. Cartan initiated and developed the external differential form and the moving frame method, and established the relationship between Lie groups and Riemannian geometry, thus laying an important foundation for the development of Riemannian geometry and opening up a broad garden with far-reaching influence. Therefore, the research of linear connection and fiber bundle has been developed.

[Name] Albert Einstein (Jewish theoretical physicist)

19 15 years, a Einstein established a new theory of gravity-general relativity by using Riemann geometry and tensor analysis tools. Riemannian geometry (strictly speaking, Lorentz geometry) and its operation method (Ritchie algorithm) have become effective mathematical tools for studying general relativity. In recent years, the development of relativity has been strongly influenced by global differential geometry. For example, vector bundle and connection theory constitute the mathematical basis of gauge field (Yang-Mills field).

In 1944, the intrinsic proof of Gauss-Bonne formula of n-dimensional Riemannian manifold is given, and his research on the characteristic class of Hermite manifold is introduced, which is later commonly known as Chen characteristic class, providing an indispensable tool for large-scale differential geometry and creating a precedent for the study of differential geometry and topology of complex manifolds. For more than half a century, the study of Riemannian geometry has developed from local to whole, and has produced many profound achievements. Riemannian geometry, partial differential equations, the theory of multiple complex variables, algebraic topology and other disciplines permeate and influence each other, which plays an important role in modern mathematics and theoretical physics.