Riemann was the first person to expand manifold into high-dimensional space. The name of manifold comes from Riemann's original German term Mannigfaltigkeit, which was translated by william king Clifford as "manifold". In his inaugural address of Gottingen, Riemann pointed out that all the values that an attribute can take constitute a Mannigfaltigkeit. He distinguished stetige Mannigfaltigkeit and discrete[ Mannigfeltikit (continuous manifold and discontinuous manifold) according to whether the changes of values are continuous or not. As an example of stetige Mannigfaltikeiten, he mentioned the color and position of an object in space and the possible shape of a space shape. He constructed an n fach ausgedehnte mannigfeltigkeit (n-expansion or n-dimensional manifold) as a continuous (n-1) fach ausgedehnte mannigfeltigkeiten heap. The concept of Mannigfaltigkeit in Riemannian intuition has developed into today's formal manifold. Riemannian manifold and Riemannian surface are named after him.
The concept of commutative cluster has been implicitly used as a complex manifold in Riemann era. Lagrange mechanics and Hamilton mechanics are essentially manifold theories in geometric sense.
Poincare studied three-dimensional manifolds and put forward a question, now called Poincare conjecture: Do all closed and simply connected three-dimensional flows have the same shape as a three-dimensional sphere? This problem has been completely solved, and the most important work is done by Russian math geek grigory perelman. China mathematician Zhu Xiping and Cao Huaidong participated in the final capping proof.
HermannWeyl gave an internal definition of differential manifold in 19 12. The basic aspects of this theme were expounded by hassler whitney and others in A.D. 1930 by using the precise intuition that began to develop in the second half of A.D. 19 century, and developed by differential geometry and Lie group theory.