1 ancient geometry
Geometry is an ancient subject, founded by the master of geometry Euclid in BC, and its power has not diminished for more than two thousand years.
Euclidean geometry is a beautiful system of axioms, it only needs to set a few simple, intuitive, universally recognized, self-evident propositions (called axioms or postulates), and then proceed from these propositions to derive Proving other propositions, and then proving more propositions by derivation, and so on, a mathematical theory is established. This is like building a high-rise building. The "axiom" is a large "brick" placed horizontally on the first floor of the foundation. With a solid and solid foundation, other bricks can be layered on top of each other, and a ten-foot-tall building can be leveled on the ground. Get up. If the base bricks are broken or placed unevenly, the building may collapse.
There are five axioms of Euclidean plane geometry. Starting from these simple five axioms, he deduced all the theorems of plane geometry and built the magnificent building of Euclidean geometry.
The miracles created by mathematical logic reasoning are astonishing. However, when people think about these axioms over and over again, they feel that the first four are obviously self-evident, and only the fifth axiom is more complicated and does not sound like a simple and easily accepted intuitive concept. So people naturally ask the question: Is this fifth axiom? Can it be derived from the other 4 axioms? What everyone means is that the building of Euclidean plane geometry may be enough to support with the front 4 "big bricks". This fifth brick was probably originally placed on top of the other four bricks.
The fifth axiom of Euclidean plane geometry, also known as the "parallel axiom", can be expressed as: "There is one and only one parallel line through a point outside the line."
A young Russian mathematician named Nikolai Lobachevsky (1792 - 1856) had a whim: if this axiom were to be changed a little, that is, a block under the What are the consequences of moving the cornerstone a little bit? For example, change it to: "Through a point outside the straight line, there are at least two parallel lines."
This change is not trivial, the difference of a few words has given birth to another geometry completely different from Euclidean geometry, which is called "non-Euclidean geometry" or "Roche geometry". The non-Euclidean edifice is equally uplifting, solid, logically complete and rigorous, but it looks a little weird.
It is inevitable that Roche's geometric system gets weird and unreasonable propositions, because the fifth postulate after being changed by Lobachevsky is not in line with people's daily life experience. How can it be possible to make multiple different straight lines that do not intersect the known straight line through a point outside the straight line on the plane? The edifice of mathematical logic built from this will of course be a monster. For example, Roche's geometry leads to the following bizarre propositions: perpendicular and oblique lines of the same line do not necessarily intersect; there is no rectangle, because all four corners of a quadrilateral cannot be right angles; there are no similar triangles; Three points do not necessarily make a circle; the sum of the three interior angles of a triangle is less than 180 degrees... What is the use of this strange "geometric building"? Some people scoffed, thinking, it's just a game played by lunatic mathematicians!
Those who laughed at Lobachevsky did not expect that decades later, non-Euclidean geometry found a place in Einstein's general theory of relativity, which is the kind of curved space described by Einstein's general theory of relativity. The geometry followed!
2 Geometrically infinitesimal
However, what is really related to the curved space of general relativity is "Riemannian geometry", which is a step further than the non-Euclidean geometry mentioned above and belongs to differential geometry.
After Euclid, Descartes invented analytic geometry, and Newton and Leibniz invented calculus. The combination of the two made the mathematics and physics of that era even more powerful and new. Using traditional axiomatic methods to study geometry, as Lobachevsky did, clearly loses out. Euler, Clello, Monge, and Gauss recognized this and created and developed differential geometry.
The pioneer of differential geometry, the French mathematician Alexis Clairaut (1713 - 1763) conducted in-depth research on the curve of space, and first studied the curvature and torsion of the curve of space (then he called it for double curvature).
What are the curvature and torsion of a curve? We recognize the curvature from the three planar curves shown in Figure 1a. The three curves in the picture are like three highways on flat ground with different shapes.
Figure 1: Curvature and torsion of the curve
We first need to introduce the tangent of a curve, or the concept of a "tangent vector". The tangent vector is the vector determined by the limit position of the connecting line when two points on the curve are infinitely close. The arrows marked on the road shown in Figure 1a are the intuitive images of the tangent vectors at each point on the curve. And what is curvature? Curvature describes how much a curve bends. For example, the top road in Figure 1a is a straight line, and the straight line will not turn. We say that its degree of curvature is 0, that is, the curvature is equal to 0. The faster the tangent vector rotates, the more the curve bends. Therefore, the curvature is mathematically defined as the rotation speed of the tangent vector of the curve to the arc length.
Curved roads on flat ground can be regarded as plane curves, and they can be described by "curvature". If the roads were built in the mountains, they had to circle up or down as they turned. At this time, the path the car travels is no longer a plane curve, but a space curve. For the road between mountains, as shown in Figure 1b, in addition to seeing the degree of bending, we can also observe the speed of the road going up (or down). We call this geometric quantity that describes the speed of the detour "torsion".
The change law of the curvature and torsion of a space curve in space completely determines the curve.
From the above study of space curves, it can be seen that the method of differential geometry is much more powerful than the method of Euclidean geometric axioms. Similar to curves, differential geometry can also be used to study surfaces, and the concepts of curvature and torsion can also be extended to surfaces to define much more complex curvature tensors.
Figure 2: Various Surfaces
The shape of the surface is infinitely variable, and according to the properties we are interested in, it can be divided into two categories: developable surfaces and non-developable surfaces. At first glance at the surfaces drawn in Figure 2a and Figure 2b, you may not see any difference between the two types of graphics. They are all shapes seen from three-dimensional space. Doesn't it look "curved" and "uneven" whether it can be developed or not? However, if you look closely, you will see that the "bending" of developable surfaces is fundamentally different from the "bending" of non-developable surfaces. Simply put, developable surfaces are "flat" in nature, they can be unfolded into a plane. For example, take the tapered surface shown in Figure 2b and cut a line with scissors to the apex, and you can lay it flat on the table without any wrinkles. Cylinders can also be cut along any line parallel to the centerline to become a plane.
但是，图2a所列举的不可展曲面，就不能展开成平面了。那是真正的、本质上的“不平”。一顶做成了近似半个球面的帽子，你无论怎样剪裁它，都无法将其没有皱褶地摊成一个平面。另一方面，你用一张平平的纸，很容易卷成一个圆筒（柱面），或者是做成一顶锥形的帽子，但你无法做出一个球面来。你顶多只能将这张纸剪成许多小纸片，粘成一个近似的球面！
谈到这儿，你大概已经基本明白了“可展”和“不可展”的区别到底是什么。尽管两类曲面在嵌入3维空间之后看起来都是弯曲的，但是，可展曲面的内在本质是“平的”，不可展曲面的内在本质是“不平”。区分这两类曲面“内在本质”的概念叫做“内蕴性”，研究这种性质的几何叫做内蕴几何。
曲面弯曲的内蕴性最早被“数学王子”高斯注意到，后为黎曼所发展，并推广到大于3的n维流形。因而，黎曼几何是一种内蕴几何。换言之，内蕴性指的是曲面（或曲线）不依赖于它在三维空间中嵌入方式的某些性质。也就是说，它是曲面某些内在的、本质的几何属性。高斯用高斯曲率——两个主曲率的乘积，来表征曲面的这种属性（图3a）。如果一个曲面的高斯曲率为0，说明它本质上是平的，是可展曲面，如图3b所示。如果一个曲面的高斯曲率不为0，说明它本质上是不平的，是不可展曲面，如图3c所示。
高斯曲率不为0的情形又有两种。正的高斯曲率对应于球面几何（图3c的下图），负的高斯曲率对应于马鞍面（图3c的上图）。马鞍面上的几何就是前面所介绍的罗巴切夫斯基几何，又被称为双曲几何。
图3：曲面的两个主曲率及高斯曲率
3爬虫的几何
可是，又该如何判定我们所面对的是哪一种几何呢？最简单的办法是测量曲面上一个三角形三个内角之和E=A+B+C。平面几何的E=180o，球面几何E>180o，双曲几何E<180o。
一个观察者在自己生活的物理空间中所能够观察和测量到的几何性质，就是这个空间的内蕴性质。比如说，球面的内蕴性质，就是生活在球面上的2维爬虫感受到的几何性质。我们人类当然是3维的生物，不是什么2维爬虫。但是，因为我们的地球很大，我们的3维尺寸比起地球来说是很小的。因此，我们可以将自己设想为某种2维生物。比如，我们在地球上测量一个大三角形，就如图3b下中的球面三角形ABC，测地员将会发现，这个三角形的三个内角都是90度，因此，内角和E=270o，大于180度。
图3b下所显示的是一个规则球面，它的空间弯曲程度到处都是一样的，但一般来说，空间的弯曲程度不一定处处相同，数学家们用“平
行移动”的概念来研究空间的弯曲程度。
什么是平行移动？简单地说，就是将一个矢量平行于自身的方向沿着空间里的一条曲线移动。像汽车上的陀螺仪那样，汽车沿公路运动时，陀螺仪总是平行于自己原来的指向。
在物理上，让大家更感兴趣的问题是：一个矢量平行移动一圈后再回到原来出发点的时候是否会有所改变？比如说，跟着汽车转了一圈的陀螺仪，指的方向是否还和原来出发时的方向一样？也许你不加思索就会给出答案：当然没有什么改变。但这是因为你习惯了用欧氏空间的直角坐标系来思考问题，从而轻易得出这个结论。如果我们假设地面是一个欧几里德平面，陀螺仪平行移动回到原处时，方向的确不会改变。但是，每个人都知道，地球是一个球体，所以我们实际上是生活在一个球面上。那么，如果从球面（或者别的曲面）的角度来研究这个问题，又会得出什么样的结论呢？
所谓“平行移动”的意思是说，在移动矢量的时候，尽可能保持矢量方向相对于自身没有旋转。好比一个女孩平行地前进、后退、左右移动，只要她的身体没有扭动，就叫平行移动。这样，当她移动一周回到出发点的时候，她认为她应该和原来出发时面对着同样的方向。如果她是在平面上移动的话，她的这个想法是正确的。但是，假如她是在球面上移动的话，她将发现自己面朝的方向可能不一样了！出发时她的脸朝左，回来时却是脸朝前，如图4b。
假如将女孩面对的方向用一个箭头（矢量）来表示。图4a所示的是一个矢量在莫比乌斯带上的平行移动，当矢量从位置1出发，沿着数字1、2、3……一直移动到10，也就是回到原来的出发位置时，得到的矢量和原来的反向。图4b中所示是球面上的平行移动，当矢量从位置1出发，沿着数字1、2、3……一直移动到7，也就是回到原来的出发位置时，得到的矢量和原来的矢量垂直。
图4：（a）莫比乌斯带上的平行移动（b）球面上的平行移动
上面的两个例子说明，矢量在曲面上平行移动一周之后，不一定还能保持原来的方向，可能与出发时有所差别。这个差别正好与曲面的高斯曲率有关系，反映了曲面内在的弯曲程度。
4阿扁的世界
下面，我们研究锥面上的平行移动，看看锥面与“真正的平面”有何不同。让我们想象有一个极小极扁的平面生物“阿扁”，生活在一张平坦的纸上，如图5a。阿扁使用直角坐标系对他的平坦世界进行观察和测量。他感受到的几何，是标准的欧几里德几何：三角形的三个内角之和等于180度；过不在同一直线上的三点，可以作一个圆；直角三角形的两条直角边的长度的平方和等于斜边长的平方……
阿扁学过微积分，还会计算许多图形的面积。阿扁经常在他的平坦世界中驾车旅行，绕行一圈回来之后，他车上的陀螺仪方向总是与原来方向相同，如图5a所示的那样。
有一天，来了一个3维世界的小生物“阿三”。阿三看中了阿扁生活的这张纸，并且灵感突发，把这张纸剪去了一个角。比如说，像图5b中图所画的情形，剪去了一个45度的角，然后将剩余图形的两条剪缝黏在一块儿，做成了一个图5c所示的锥面。阿扁是个2维小爬虫，他看不见阿三，也感觉不到阿三的存在，更不可能知道阿三对他的世界干了些什么。
不过，生活在纸上的阿扁并没有立即感到他的世界有什么变化。照样是欧氏几何，他画的直角坐标轴仍然在那儿。当他拿着他的（平面）陀螺仪，沿着他的小圆圈（如图5b中的C1那样）旅行，进而回到原来出发点的时候，陀螺仪的指向和原来一样。这说明，矢量平行移动的规律好像没有任何改变。
图5：阿扁的世界
阿扁的技术越来越高，胆子越来越大，旅游路线也走得越来越远。他逐渐发现了一些问题。比如说，当他沿着图5b中所示的C2那样的曲线走一圈回到原出发点时，他的陀螺仪的指向和出发时有了一个45度的角度差。这个新发现令阿扁既激动又困惑。于是，他进行了更多的带陀螺仪绕圈实验，绕了好多个不同的圈，终于总结出了一个规律：他生活的世界中，在右图中所标记的点O附近，是一个特殊的区域，只要他移动的闭曲线中包含了这个区域，陀螺仪的指向就总是和原来出发时的方向相差45度左右。如果行走的圈没有包括这个点的话，便不会使陀螺仪的方向发生任何改变。当时的阿扁，技术还不够精确，还没有搞清楚这个区域是多大，况且，他也有点害怕那块神秘兮兮的地方，不敢在那儿逗留过久，作太多的探索，以防遭遇生命危险。
阿扁喜欢读书学习新知识，他从一本数学书中了解到，如果陀螺仪走一圈方向改变的话，说明你所在的空间是弯曲的。因此，通过对多次实验结果的总结归纳，阿扁提出一个假设：他所在的世界基本是平坦的，除了那块该死的区域之外！
再回到我们的世界来看待球面几何。陀螺仪走一圈后方向改变的值，叫做平行移动一周后产生的角度亏损，可用θ表示。角度亏损与空间的高斯曲率有关，一个标准球面上的高斯曲率处处相等。因此，如果有某种生活在球面上的扁平生物的话，他沿任何曲线绕行一圈后，陀螺仪方向都会有变化，而且，角度亏损θ是不固定的，它与绕行回路所包围的球面面积A成正比，其比例系数对球面而言是一个定植，就等于曲面的高斯曲率α。角度亏损θ = α*A。
如果研究对象不是标准的球面，而是一般的2维曲面，上述“角度亏损θ正比于区域面积A”的结论在大范围内不能成立，但在2维曲面某个给定的P点附近，当绕行的回路趋近于无限小的时候仍然成立。也就是说：无限小的角度亏损dθ将正比于无限小的区域面积dA：dθ = α*dA。这时的α= dθ/dA，便是曲面上这一点的曲率。
阿扁也想通了这些道理，明白他的世界不是球面，而大多数地方都是平面，只有一点不对，那一点附近的空间是弯曲的。
Ah Bian applies the above relationship between the curvature of the surface and the loss of the infinitesimal parallel translation angle (α = dθ/dA) to the conical surface. Because the cone is a developable surface. Its geometry everywhere is the same as Euclidean geometry on the plane, except for that vertex. That is, every point on the cone has a curvature equal to 0, but the vertex is a singularity with a curvature equal to infinity.
With this mathematical knowledge, Ah Bian suddenly realized: So the world I live in is a cone!
Figure 6: Parallel movement on cone and sphere
Humans are creatures in 3-dimensional space, and our world is 3-dimensional. Just like the "A San" described earlier, of course, he is much smarter than the poor flat creature "A Bian". Ah Bian repeated the measurements many times, and used his 2-dimensional flat head to make extremely difficult "abstractions", and only then did he understand his cone world! And when we look at 2D in the 3D world, we can see it very clearly: the cone surface is a developable surface, or in other words, it was originally made by Asan cut a corner of a flat "paper" and glued it. of. Therefore, we can see at a glance that Ah Bian's cone world is flat everywhere, except for that one vertex O.
When doing parallel movement on a cone, why is there an angular loss when the movement path includes vertex O? This problem is easier to understand from our 3D world. In Figure 6a, we cut the cone from the vertex and re-unfold it into a flat shape. The difference between this "plane figure with one corner cut off" and the whole Euclidean plane is that A and B in the figure are the same point on the cone, so the straight lines OA and OB need to be understood as the same line. In this way, we understand the source of the angle loss.
∑ Editor | Gemini
Source | Science Network blogger article / Zhang Tianrong
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