Geometry is the study of spatial structure and properties.

It is one of the most basic research contents in mathematics, which has the same important position as analysis and algebra, and has a very close relationship.

Development and branches of geometry

Geometric development

Geometry has a long history and rich contents. It is closely related to algebra, analysis and number theory. Geometric thought is the most important thought in mathematics. At present, the development of all branches of mathematics tends to be geometric, that is, to explore various mathematical theories with geometric viewpoints and thinking methods.

Plane geometry and solid geometry

The earliest geometry belongs to plane geometry. Plane geometry is to study the geometric structure and measurement properties (area, length, angle) of straight lines and quadratic curves (that is, conic curves, that is, ellipses, hyperbolas and parabolas) on the plane. Plane geometry adopts axiomatic method, which is of great significance in the history of mathematical thought.

The content of plane geometry naturally transferred to solid geometry of three-dimensional space. In order to calculate the problem of volume and area, people have actually started to involve the original concept of calculus.

After Descartes introduced the coordinate system, the relationship between algebra and geometry became clear and increasingly close. This prompted the emergence of analytic geometry. Analytic geometry was independently founded by Descartes and Fermat. This is another landmark event. From the perspective of analytic geometry, the properties of geometric figures can be attributed to the analytical properties and algebraic properties of equations. The classification of geometric figures (such as dividing conic curves into three categories) is transformed into the classification of algebraic characteristics of equations, that is, the problem of finding algebraic invariants.

Solid geometry comes down to the research category of analytic geometry in three-dimensional space, so the study of geometric classification of quadric surfaces (such as sphere, ellipsoid, cone, hyperboloid and saddle surface) comes down to the study of quadratic invariants in algebra.

Generally speaking, the above geometry is investigated under the background of the geometric structure of Euclidean space, that is, the plane space structure, and does not really pay attention to the geometric structure of surface space. Euclid's geometric axiom essentially describes the geometric characteristics of flat space, especially the fifth postulate has aroused people's doubts about its correctness. Therefore, people began to pay attention to the geometry of its curved space, that is, "non-Euclidean geometry". Non-Euclidean geometry includes several classical geometric topics, such as "spherical geometry" and "Roche geometry". On the other hand, people began to consider projective geometry in order to introduce those elusive points at infinity into the observation range.

Generally speaking, these early non-Euclidean geometries studied non-metric properties, that is, they had little to do with metrics, and only paid attention to the positions of geometric objects, such as parallelism, intersection and so on. The spatial background studied by these kinds of geometry is a curved space.

differential geometry

In order to introduce the theoretical measurement (length, area, etc. ), we need to introduce calculus to analyze the properties of space bending locally. Differential geometry came into being. The study of differential geometry of curves and surfaces is called classical differential geometry. The objects discussed in classical differential geometry must be embedded in Euclidean space before defining various geometric concepts (such as tangent and curvature). If a geometric concept has nothing to do with the spatial position of geometric objects, but only with its own state, it is said to be inherent. In physical language, geometric properties must be independent of the choice of reference system.

Intrinsic geometry

What geometric concepts are inherent? This was the most important theoretical problem at that time. Gauss found that the curvature of a surface (that is, the quantity reflecting the degree of bending) is actually intrinsic-although its initial definition seems to be related to its large spatial position. This important discovery is called Gauss's wonderful theorem. Another important discovery of classical geometry is Gauss-Bohnert formula, which reflects the relationship between curvature and sum of triangles in curved space.

Riemannian geometry, the first discipline to study intrinsic geometry, created this basic theory in a famous speech. It emphasizes the intrinsic thought for the first time, classifies all geometric objects in the past into a more general category, and internally defines geometric concepts such as measurement. This geometric theory has opened the door to modern geometry and is of milestone significance. It also became the mathematical basis of Einstein's general theory of relativity.

Since Riemannian geometry, differential geometry has entered a new era, and geometric objects have expanded to manifold (a curved geometric object)-the concept introduced by Poincare. From this, tensor geometry, Riemann surface theory, complex geometry, Hodge theory, fiber bundle theory, Finsler geometry, Morse theory, deformation theory and so on have been developed.

From the angle of algebra, geometry has developed from traditional analytic geometry to a more general theory-algebraic geometry. Traditional algebraic geometry takes the zero set of polynomial equations as the geometric object to study the geometric structure and properties-this kind of geometry is called algebraic cluster. The straight lines, conic curves, spheres and cones studied in analytic geometry are all special cases. A little generalization is algebraic curve, especially plane algebraic curve, which corresponds to Riemannian surface. Algebraic geometry can be described by the language of commutative algebraic rings and modules, and can also be discussed by complex geometry and Hodge theory. The idea of algebraic geometry has also been introduced into number theory, thus promoting the development of abstract algebraic geometry, such as arithmetic algebraic geometry.

analysis situs

Topology is an important subject closely related to traditional geometry. It can also be regarded as a kind of "flexible" geometry, and it is also the research basis of all geometry. The rudiment of this subject was founded by Poincare and later developed into a mature mathematical theory. Topology is the core of mathematics. Some basic characteristics of describing geometric objects, such as genus (number of holes), are discussed. From this, homology theory, homotopy theory and other basic theories were developed.

Other geometric disciplines

In addition to the above traditional geometry, we also have Minkowski's Geometry of Numbers. Tropical geometry, a new discipline closely related to modern physics; Discuss fractal geometry of dimension theory; There are also convex geometry, combinatorial geometry, computational geometry, permutation geometry, intuitive geometry and so on.