First, the establishment of Euclidean geometry.
Euclidean geometry is the abbreviation of Euclidean geometry, and its founder is Euclid, a great mathematician in ancient Greece in the third century BC. Before him, the ancient Greeks had accumulated a lot of geometric knowledge and began to use logical reasoning to prove the conclusions of some geometric propositions. Euclid, a great geometric architect, arranged the geometric propositions according to the logical system on the basis of the "wood, stone and brick" materials prepared by predecessors, built a towering geometric building and completed the glorious work "Geometry Elements" in the history of mathematics. The publication of this book marks the establishment of Euclidean geometry. This scientific work is the most widely distributed and used book. Later, it was translated into many languages, with more than 2000 versions. Its appearance is an extremely far-reaching event in the history of mathematics development and a milestone in the history of human civilization. For more than two thousand years, this book has been occupying the leading position in geometry teaching, and its position has not been shaken so far. Many countries, including China, still use it as a geometry textbook.
Second, the immortal monument.
Euclid sorted out many early theorems that were unrelated and not strictly proved, and wrote the book Geometry Elements, which turned geometry into an immortal monument based on logical reasoning. This epoch-making work is divided into 13 volumes and 465 propositions. There are eight volumes about geometry, including plane geometry and solid geometry. However, the significance of "Elements of Geometry" is by no means limited to the importance of its contents or its excellent proof of theorems. What really matters is a method called axiomatization that Euclid created in his book.
When proving geometric propositions, each proposition is always derived from the previous proposition, and the previous proposition is derived from the previous proposition. We can't deduce this indefinitely, but we should start with some propositions. These propositions, which are self-evident and recognized as the starting point of argument, are called axioms, such as "two points determine a straight line" that students have learned. Similarly, there are some undefined primitive concepts, such as points and lines. In a mathematical theoretical system, we take as few original concepts and some unproven axioms as possible, and build the system into a deductive system by pure logical reasoning. This method is the axiomatic method. This is the method that Euclid adopted. He first laid out axioms, postulates and definitions, and then systematically proved a series of propositions from simple to complex. Taking axioms, postulates and definitions as elements, he first proved that the first proposition was known. Then prove the second proposition on this basis, and so on, prove a large number of propositions. Its wonderful argument, meticulous logic and rigorous structure are amazing. Scattered mathematical theories were successfully woven into a system from basic assumptions to the most complicated conclusions by him. Therefore, in the history of mathematical development, Euclid is considered to be the first person to successfully and systematically apply axiomatic methods, and his work is recognized as the first model to establish a deductive mathematical system by axiomatic methods. It is in this sense that Euclid's Elements of Geometry has had a great and far-reaching impact on the development of mathematics and set up an immortal monument in the history of mathematical development.
Third, the perfection of Euclidean geometry.
Axiomatic method has penetrated into almost every field of mathematics, which has had an immeasurable impact on the development of mathematics. Axiomatic structure has become the main feature of modern mathematics. As the earliest model of axiomatic structure, geometric elements still have many shortcomings in logical rigor by modern standards. For example, an axiomatic system has some primitive concepts (or undefined concepts), such as points, lines and surfaces. Euclid defined all these, but the definition itself is vague. In addition, its axiomatic system is not complete, and many proofs have to rely on intuition. In addition, individual axioms are not independent, that is, they can be deduced from other axioms. These defects were perfected in 1899 when the German mathematician Hilbert published his Basic Geometry. In this masterpiece, Hilbert successfully established a complete and rigorous Euclidean geometry axiom system, the so-called Hilbert axiom system. The establishment of this system makes Euclidean geometry a very perfect and rigorous geometric system with logical structure. It also marks the end of Euclidean geometry perfection.
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Riemann geometry
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.
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Roche geometry
The only difference between the axiom system of Rothschild geometry and Euclid geometry is that a pair of scattered straight lines in Europe are infinitely far away from the geometric parallel axiom on both sides of their only common vertical line, instead of "starting from a point outside the straight line, at least two straight lines can be parallel to this straight line", and other axioms are basically the same. Due to the difference of parallel axioms, a series of new geometric propositions different from Euclidean geometry are derived through deductive reasoning.
As we know, Rothschild geometry adopts all the axioms of European geometry except one parallel axiom. Therefore, any geometric proposition that does not involve parallel axioms, if correct in Euclidean geometry, is also correct in Rothschild geometry. In Euclidean geometry, all propositions involving parallel axioms are not valid in Ronaldinho geometry, and they all have new meanings accordingly. Here are a few examples to illustrate:
Euclidean geometry:
The perpendicular and diagonal of the same line intersect.
Two straight lines perpendicular to the same straight line or parallel.
There are similar polygons.
Crossing three points that are not in a straight line can be done, and only a circle can be made.
Roxburgh geometry
The perpendicular and diagonal of the same line do not necessarily intersect.
Two straight lines perpendicular to the same straight line spread to infinity when both ends are extended.
There are no similar polygons.
Passing three points that are not on the same straight line may not necessarily make a circle.
It can be seen from the above propositions of Cartesian geometry that these propositions are contradictory to the intuitive image we are used to. Therefore, some geometric facts in Rothschild geometry are not as easily accepted as European geometry. But mathematicians are right to suggest that we can use the facts in Euclidean geometry we are used to as an intuitive "model" to explain Rothschild geometry.
1868, Italian mathematician Bertrami published a famous paper "An Attempt to Explain Non-Euclidean Geometry", which proved that non-Euclidean geometry can be realized on the surface of Euclidean space (such as quasi-sphere). In other words, non-Euclidean geometry propositions can be translated into corresponding Euclidean geometry propositions. If there is no contradiction in Euclidean geometry, there is no contradiction in non-Euclidean geometry.
Since people admit that Euclidean geometry is not contradictory, it is natural to admit that non-Euclidean geometry is not contradictory. Until then, non-Euclidean geometry, which has been neglected for a long time, began to get extensive attention and in-depth research in academic circles, while Lobachevsky's original research was highly praised and praised by academic circles, and he himself was also known as "Copernicus in geometry".