As early as when Gauss was fifteen years old, he conceived a geometry, in which the fifth postulate of Euclidean geometry was no longer valid. He called this geometry "starry sky geometry", and perhaps he predicted that this geometry might be realized in the vast starry sky.

But as we all know, it is Lobachevski who really put forward this geometry openly and systematically (some English documents are Lobachevsky, and Russian names can be translated into English with a slight difference). ) So this geometry is called "Lobachevski geometry", also known as hyperbolic geometry. In hyperbolic geometry, the sum of the internal angles of a triangle is no longer equal to 180 degrees. But what we need is not only the qualitative result, but also the degree of deviation between the sum of internal angles and 180 degrees, which is called "angular surplus" and angular surplus. Of course, this surplus is a surplus in a broad sense. If the difference is negative, it is a negative surplus:)

This difference is described by the famous (local) Gauss-Bonne theorem, which directly relates the curvature of the surface to the angular surplus. The sum of the integral of Gaussian curvature K of a polygon on a surface plus the integral of Geodesic curvature k_g of a polygon boundary curve on the boundary plus the polygon external angle is equal to 2π. If the boundary curve of this polygon is geodesic, then Geodesic curvature is 0, and Geodesic curvature's integral is 0, so the calculation will be greatly simplified. If it is a geodesic triangle, then we can immediately get the generalization of the formula of the interior angle of the triangle. Due to the complementary relationship between the inner angle and the outer angle, the formula will become: the sum of the inner angles of a triangle MINUS π equals the integral of Gaussian curvature K on the surface surrounded by the triangle. So we can know:

If k is equal to zero, then this is just a plane triangle, and the remainder of the angle is zero, and the sum of the internal angles of the triangle is equal to π;

If k is greater than zero, it is similar to a triangle on a sphere with positive angular surplus, and the sum of the internal angles of the triangle is greater than π;

If k is less than zero, then it is similar to a triangle on a pseudo sphere, with a negative angular surplus, and the sum of the internal angles of the triangle is less than π.

Therefore, even if Gauss-Bonnet formula is specialized twice (the first time is that the Geodesic curvature of the polygon boundary curve is zero, and the second time is that the polygon is a triangle), these three beautiful results can still be obtained, which directly generalizes the formula of triangle interior angle sum.

The Gauss-Bonnet theorem of the whole is more beautiful: the integral value of Gauss curvature of compactly oriented two-dimensional Riemannian manifold M (which can be roughly regarded as a generalization of a surface) is equal to 2ππ(M), where χ(M) is the Euler characteristic number of m, which is a typical discrete value of the whole, and the Gauss curvature can take local values continuously. Here, Geodesic curvature's line integral was cancelled directly. Let's think about the beautiful result that the auxiliary line integrals cancel each other when the complex variable function proves the Cauchy integral theorem of the multi-connected domain (in fact, we have this method when we prove the Glenn theorem of the multi-connected domain), and we can imagine this result by analogy. Only in the proof of global Gauss-Bonet theorem, the famous "triangulation" is used to divide the region into triangles, which counteracts the line integral (triangulation is also used in the modern proof of Cauchy integral theorem for simply connected regions), while in Cauchy integral theorem for multi-connected regions, the multi-connected regions are divided into simply connected regions. It can also be seen from here that the research in many fields of mathematics has the same effect. This formula skillfully combines two different important concepts perfectly.

Later, curvature became the core concept in geometry after Riemann generalization, and Euler characteristic number became the core concept in topology after Poincare generalization. These two concepts have been skillfully combined in global differential geometry, and this ingenious combination is due to Chen Shengshen's direct and internal promotion of Gauss-Bonnet theorem on high-dimensional complex manifolds. Sure enough, the sentence "the dragon gives birth to the dragon, the phoenix gives birth to the phoenix, and the mouse son can make holes" was answered. Great theorems, after great promotion, will produce even greater disciplines.

When Weil and Allendorff used block cutting to embed in high-dimensional Euclidean space to prove the generalization of this theorem, Nash embedding theorem had not yet appeared, so the premise was not established from the beginning. It is really not satisfactory to increase an internal good result, but to improve it in an external way. So as soon as Chen Shengshen arrived in the United States, Weil told Chen Shengshen about this idea and came to the conclusion that there must be an inherent proof method for this theorem. Chen Shengshen quickly completed the proof. Weyl, one of the best mathematicians at that time, was surprised by this result and praised it greatly. Weil concluded that this is a great work with geometric milestone significance.

Here, from hyperbolic geometry to the famous Gauss-Bonne-Chen theorem, we also mention a person, that is, the great Riemann, who founded Riemann geometry in a narrow sense. Then, this result is incorporated into his extremely profound Riemannian Geometry (the difference between Riemannian Geometry and Riemannian Geometry is clear, their formal difference is "ian", and the substantive difference is the difference between "constant curvature" and "arbitrary curvature"), which summarizes the intrinsic geometry of Gaussian surfaces. The abstract Riemann metric is defined, which directly gets rid of the embedding research in Euclidean space only in two dimensions, so that the surface research is no longer equivalent to the surface research in three-dimensional Euclidean space. Poincare metric is defined on the famous Poincare upper plane and cannot be embedded in three-dimensional Euclidean space. Poincare metric is one of Riemann metrics.

As Milnor said, before Riemannian geometry appeared, hyperbolic geometry was just a trunk without hands and feet. Riemann made this torso normal.

After Riemann, beltrami realized the local hyperbolic geometry on the pseudo sphere, Klein realized the global hyperbolic geometry on the open unit circle (excluding the circumference), and Poincare realized the global hyperbolic geometry on the upper half plane (excluding the real axis). It is easy to prove that there is a * * shape mapping between the unit circle and the upper half plane, and the unit circle and the real number axis are also in one-to-one correspondence as the boundary of the two fields. Given the Poincare metric on the unit circle, the curvature of its section can be calculated as-1, which proves that the spatial curvature of hyperbolic geometry is less than zero. As we know, hyperbolic geometry was developed after Poincare's death, and the most outstanding figure was thurston, winner of Fields Prize. In addition, the development of this subject is very slow, which shows that it is difficult and Poincare is great.

As early as the age of 26, the famous Schwarzenegger considered what the radius of curvature should be if the universe was curved. At the end of 19, he said: "In this century, some people put forward non-Euclidean geometry except Euclidean geometry, and the main examples are spherical space and pseudo-spherical space. If we know what the world looks like in spherical and pseudo-spherical geometry, we will be surprised that these geometries may have limited radius of curvature. If this is possible, you will feel that you are in a geometric wonderland; Whether such a wonderful fairyland will become a reality, we have no way of knowing. " (Excerpted from 1986, words quoted by Chandraseka in Schwarzenegger's lecture, translated by Yang Jianye, Wang Xiaoming, etc. )

He also estimated the curvature radius limit of three-dimensional space by using astronomical data at that time, and thought that the lower limits of curvature radius of hyperbolic space and spherical space were 64 light years and 1600 light years respectively.

Of course, we know that in 1900, astronomical ranging technology is still not perfect. In fact, when Einstein put forward the static cosmological model (19 17), his understanding of the size of the universe was still very vague. Even when Hubble put forward the expansion theory of the universe, the observation of the universe was seriously wrong because of the analysis of the luminosity of Cepheid variable stars. So it's really amazing to have such dreams and calculations about the universe in the Swahili era. His thoughts have penetrated into hyperbolic geometry and elliptic geometry.

Beyond the point, the difficulties faced by modern differential geometricians in dealing with three-dimensional problems and four-dimensional problems are completely different, because if Ricci curvature in three-dimensional space is zero, Riemann section curvature is zero, while four-dimensional space does not have this property. But in Schwarzenegger's time, he certainly didn't consider this, so if he was smart enough to directly consider four-dimensional space-time, he would still go into battle with his sword:)

We also know that when Lobachevski put forward hyperbolic geometry, he imagined that hyperbolic geometry might be realized in the universe. He said: "At the same time, we can't help but pay attention to Laplace's point of view: the stars we see are only a part of the celestial body, just like dim, looming spots, similar to what we see in constellations such as Orion and Capricorn. Therefore, not to mention the infinite extension of space in imagination, the distance that nature itself shows us is insignificant even compared with the distance from our earth to the stars. In addition, it cannot be further asserted that assuming that the measurement of a straight line does not depend on the angle-this assumption, which many geometricians want to adopt as a strict truth, has not been proved-may find that it has a detectable error before we transition to the limit of the visible world. "

This problem was actually imagined by Clifford in England, but the dream was further deepened by Schwarzenegger. In this way, we can understand why Schwarzenegger gave the first exact solution as soon as Einstein worked out the general theory of relativity. I'm an old hand. It's easy to learn these new geometries. Coupled with his special ability to solve partial differential equations, Einstein admired this result, which was more sincere than Friedman's treatment six years later.

We should also talk more about elliptic geometry, because it is non-Euclidean geometry like hyperbolic geometry, but considering the substantial leap from Euclidean geometry to hyperbolic geometry, the leap from hyperbolic geometry to elliptic geometry is almost zero. It's just a parallel development. I didn't mean to belittle Riemann. Elliptic geometry is the "narrow Riemannian geometry" mentioned above. With the generalized Riemann geometry, Riemann's greatness no longer needs this consolation prize, not to mention many other supreme honors: Riemann surface, Riemann hypothesis and so on.

At the end of the article, I remembered a coincidence, that is, Gauss and Schwarzenegger both served as curators of the Gottingen Planetarium. One is astronomy because of mathematics, and the other is mathematics because of astronomy. That's great.