A branch of mathematics.

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Differential geometry is a branch of mathematics that uses calculus theory to study the geometric properties of space. Classical differential geometry studies curves and surfaces in three-dimensional space, while modern differential geometry begins to study manifold, a more general space. Differential geometry is closely related to other branches of mathematics such as topology and has an important influence on the development of physics. Einstein's general theory of relativity is based on Riemann geometry in differential geometry.

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The history of differential geometry

origin

The emergence and development of differential geometry is closely related to calculus. The first person who made a contribution in this respect was the Swiss mathematician L.Euler. In 1736, he first introduced the concept of intrinsic coordinates of plane curves, that is, the geometric quantity of the arc length of curves was taken as the coordinates of points on the curves, and thus began to study the intrinsic geometry of curves. /kloc-at the beginning of the 9th century, gaspard monge, a French mathematician, first applied calculus to the study of curves and surfaces, and published the book Analysis in Geometry in 1807, 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.

develop

1827, the German mathematician 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 surface theory. Gauss mastered the most important concepts and basic contents in differential geometry and established the intrinsic 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 curve on the surface, the included angle between two curves, the area of a certain area on the surface, geodesic, Geodesic curvature and total curvature.

1854, German mathematician B. Riemann extended Gauss's theory to N-dimensional space in his inaugural speech, which was the birth of Riemann geometry. Later, many mathematicians, including E. beltrami, E. B. Christophel, R. Lipshits, L. Bianchi and T. Matteo Ricci, began to study along Riemann's ideas. Among them, Bianchi was the first author with the topic of "Differential Geometry".

1870, when the German mathematician Felix Klein delivered his inaugural speech at the University of Herun, Germany, he expounded his Herun Root Program and classified the existing geometry with transformation groups. In the half century after the publication of Herun Root Program, it became the guiding principle of geometry, promoted the development of geometry, and led to the establishment of projective differential geometry, affine differential geometry and * * differential geometry. Especially the projective differential geometry, began in 1878 Alfan's paper, 1906 American school represented by Wilsinsky, 19 16 Italian school headed by Fubini. In affine differential geometry, blaschke has also done decisive work.

Global differential geometry

French mathematician E Catan emphasized the concept of connection in differential geometry and established the concept of external differential. This is the basic work of global differential geometry. Subsequently, China mathematician Chen Shengshen extended Gauss-Bonet theorem to surfaces from the viewpoint of external differential. Since then, differential geometry has become an indispensable field of modern mathematics.

basic content

Differential geometry takes smooth curves (surfaces) as the research object, so the whole differential geometry is developed from the concepts of arc length of curves and tangent of a point on curves. Because differential geometry studies the properties of general curves and surfaces, the curvature of a plane curve at one point and the curvature of a space curve at one point are important discussions in differential geometry, and it is necessary to calculate the curvature of each point on a curve or surface by differential method.

There are two important concepts on the surface, namely, distance and angle on the surface. For example, there are countless paths from one point to another on a surface, but there is only one shortest path between these two points, which is called geodesic from one point to another. In differential geometry, we should discuss how to judge the curve on a surface as the geodesic of this surface, and also discuss the properties of geodesic. In addition, discussing the curvature of a surface at each point is also an important content of differential geometry.

differential geometry

In differential geometry, the so-called "active scalar method" is often used to discuss the properties of the neighborhood of each point on an arbitrary curve. To study the "small range" property of any curve, we can also "transform" this curve into an elementary curve through topological transformation.

In differential geometry, due to the application of mathematical analysis theory, high-order infinitesimal can be omitted in an infinitesimal range, some complex dependencies can become linear, and uneven processes can also become uniform. These are unique research methods of differential geometry.

Application and influence

In modern times, differential geometry is closely related to topology, variational theory, Lie group theory and so on because of studying the global properties of curves and surfaces in high-dimensional space. These fields of mathematics and differential geometry permeate each other and become one of the central topics of modern mathematics.

Differential geometry is widely used in mechanics and some engineering problems, for example, in the application of elastic thin shell structure and mechanical gear meshing theory, differential geometry theory has been fully applied.

The study of differential geometry has an inestimable influence on other branches of mathematics, mechanics, physics and engineering. For example, the geometry on the pseudosphere is closely related to non-Euclidean geometry; Geodesic is deeply related to mechanics, variational method and topology, and it is a rich research topic. In this field, the research results led by J. Adama, H. Poincare and others are quite abundant. Minimal surface is a research field with deep connection with complex variable function theory, variational theory and topology, and K. Weierstrass and J. Douglas have made outstanding contributions.

differential geometry