Geometry on riemannian manifolds. Geometric theory put forward by German mathematician G.F.B Riemann in the middle of the 9th century. 1854, Riemann's inaugural speech entitled "On Hypothesis as the Basis of Geometry" published at the University of G? ttingen is generally considered as the source of Riemann's geometry. In this lecture, Riemann regards the surface itself as an independent geometric entity, not just a geometric entity in Euclidean space. He first developed the concept of space, and proposed that the object of geometry research should be a multiple generalized quantity, and the points in space can be described by n real numbers (x 1, ..., xn) as coordinates. This is the original form of modern n-dimensional differential manifold, which lays the foundation for describing natural phenomena in abstract space. Geometry in this space should be based on the distances between two infinite adjacent points (x 1, x2, ... xn) and (x 1+dx 1, ... xn+dxn), and measured by the square of the differential arc length. that is

,

(gij) is a positive definite symmetric matrix composed of functions. This is the Riemann metric. The differential manifold that gives the Riemannian metric is a Riemannian manifold.

Riemann realized that a metric is just a structure added to a manifold, and there can be many different metrics on the same manifold. Mathematicians before Riemann only knew that there was an induced metric DS2 = edu2+2fdudv+gdv2 on the surface S in the three-dimensional Euclidean space E3, that is, the first basic form, but they didn't realize that S could also have a metric structure independent of three-dimensional Euclidean geometry. Riemann realized the importance of distinguishing induced metric from independent Riemann metric, thus getting rid of the bondage of classical differential geometric surface theory limited to induced metric and establishing Riemann geometry, which made outstanding contributions to the development of modern mathematics and physics.

Riemannian geometry takes Euclidean geometry and various non-Euclidean geometries as its special cases. For example, a measure (a is a constant) is defined, which is a common Euclidean geometry when a = 0, an elliptic geometry when a > 0 and a hyperbolic geometry when a < 0.

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 had not developed, so Riemann geometry was limited to a very small theoretical range. 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.

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.

Riemann conjecture, that is, the distribution of prime numbers, ultimately boils down to the following so-called Riemann zeta function:

∞ 1

ζ(z)=∑—, z = x+iy

N= 1 NZD

He put forward the conjecture that all zeros of zeta (z) function between 0 ≤ x≤ 1 are above x = 1/2, that is, the real part of zero is 1/2, which is still an unsolved problem.

The Different Functions of Riemannian Geometry and Euclid Geometry

In mathematics, Euclidean geometry still dominates; In physics, Riemann geometry is used.

Because according to Riemann geometry, light moves according to a curve; However, in Euclidean geometry, light moves in a straight line.