The emergence and development of differential geometry is closely related to mathematical analysis. Euler, a Swiss mathematician, made the first contribution in this respect. In 1736, he first introduced the concept of intrinsic coordinates of plane curves, that is, taking the arc length of curves as the coordinates of points on curves, and thus began to study the intrinsic geometry of curves.
1At the beginning of the 8th century, the French mathematician gaspard monge first applied calculus to the study of curves and surfaces, and in 1807, he published the book The Application of Analysis in Geometry, which is the earliest work of differential geometry. In these studies, we can see that the growing demands of mechanics, physics and industry are the factors that promote the development of differential geometry.
1827, Gauss published the book General Research on Surfaces, which is of great significance in the history of differential geometry, and its theory laid the foundation of modern formal surface theory. After the development of differential geometry has gone through 150 years, Gauss mastered the most important concepts and basic contents in differential geometry and established the inner geometry of surfaces. Its main idea is to emphasize some properties on the surface that only depend on the first basic form, such as the length of the surface on the surface, the included angle between two curves, the area of a region on the surface, geodesic, geodesic curvature and total curvature. His theory laid the foundation of modern formal surface theory.