内容来源：https://www.quantamagazine.org/what-is-a-manifold-20251103/

内容总结：

【本报专稿】在我们日常看来平坦的大地，实则是一个巨大的球体。这种“局部看似平坦，整体结构却可能异常复杂”的几何对象，正是数学中被称为“流形”的重要概念。这一由德国数学家黎曼于19世纪中期开创的理论，彻底革新了人类对空间本质的认知。

19世纪前，几何学研究始终局限于欧几里得平直空间。随着数学发展，学者们开始探索球面、鞍面等弯曲空间，却发现传统几何法则在此纷纷失效。1854年，原本攻读神学的黎曼在哥廷根大学发表了划时代的就职演讲，将高斯关于曲面的研究推广到任意维度。尽管当时学界认为其理论过于抽象，这项成果却在半个世纪后获得庞加莱等大家认可，更成为爱因斯坦构建广义相对论的数学基石。

流形的核心特征在于其“局部平直性”——无论圆形曲线或地球表面，在足够小的尺度下都会呈现直线或平面的特性。数学家通过构建“坐标卡集”来研究流形，如同用多张地图拼合呈现复杂地形。这种方法使研究者能运用成熟的计算工具，逐片分析后再整合全局信息。

如今，流形理论已深度融入现代科学体系：爱因斯坦用四维流形描述时空结构，工程师通过环面流形模拟双摆运动，数据科学家借其梳理高维神经网络数据。正如普林斯顿大学物理学家乔纳森·索斯所言：“流形之于科学，犹如字母之于语言——它们是一切的基础。”这项诞生于19世纪的抽象数学理论，正持续为人类探索宇宙奥秘提供关键认知框架。

中文翻译：

何为流形？

引言
站在田野中央时，我们很容易忘记自己生活在一个圆形星球上。与地球相比，人类如此渺小，以至于在我们眼中，大地是平坦的。

世界上充满了这类形状——在栖身其上的蚂蚁看来是平坦的，实则可能具有更复杂的整体结构。数学家将这类形状称为“流形”。19世纪中期，伯恩哈德·黎曼提出的流形概念彻底改变了数学家对空间的认知。空间不再只是其他数学对象的物理背景，而是本身值得研究的、精确定义的抽象对象。

这种新视角让数学家能严谨探索高维空间，从而催生了专注于研究流形等数学空间的现代拓扑学。流形还在几何学、动力系统、数据分析和物理学等领域占据核心地位。

如今，流形为数学家提供了解决各类问题的通用语言。它们之于数学，如同字母之于语言。“会西里尔字母就等于会俄语吗？”意大利比萨大学数学家法布里齐奥·比安奇举例道，“当然不。但若不识字母，又如何学俄语？”

那么流形究竟是什么？它们又提供了怎样的语言工具？

思想成形
数千年来，几何学始终意味着对欧几里得空间——即我们周遭的平坦空间——中物体的研究。“19世纪前，‘空间’即指‘物理空间’，”西班牙塞维利亚大学科学哲学家何塞·费雷罗斯解释道，它就像一维的直线或二维的平面。

在欧几里得空间中，万物遵循直觉：两点间最短距离是直线，三角形内角和恒为180度，微积分工具可靠且定义明确。

但到19世纪初，部分数学家开始探索非平坦的弯曲几何空间，如球面或鞍形曲面。在这些空间里，平行线可能相交，三角形内角和可能大于或小于180度，微积分运算也变得复杂得多。

当时的数学界难以接受（甚至理解）这种几何思维的变革。

但有些数学家希望更进一步，性格腼腆的黎曼便是其中之一。他原计划继承牧师父亲的衣钵研读神学，最终却被数学吸引。1849年，他选择师从卡尔·弗里德里希·高斯攻读博士，后者当时正研究独立于外围空间的曲线与曲面内在性质。

1854年，为获得哥廷根大学教职，黎曼被迫进行公开演讲。指定题材是几何学基础。尽管畏惧公开演说，他仍在6月10日阐述了将高斯曲面几何思想推广至任意维度（甚至无限维）的新理论。

这场融合数学、哲学与物理学的演讲令高斯当即叹服。但多数数学家认为黎曼的理念过于模糊抽象，缺乏实用价值。费雷罗斯指出：“许多科学家和哲学家斥之为‘无稽之谈’。”因此数十年来，这项成果备受冷遇。黎曼的讲稿直到他去世两年后的1868年才正式出版。

但至19世纪末，亨利·庞加莱等数学巨擘已认识到黎曼思想的重要性。1915年，阿尔伯特·爱因斯坦将其运用于广义相对论，使该理论从哲学抽象走向现实世界。到20世纪中期，它已成为数学的基础支柱。

黎曼提出的这个概念能囊括所有维度的几何形态，彻底改变了数学家看待空间的方式。

这就是流形。

测绘版图
“流形”一词源于德语“Mannigfaltigkeit”，意为“多样性”或“多重性”。

流形具有这样的特性：在其任意点放大观察，局部都呈现欧几里得空间特征。例如圆是一维流形，任意位置放大都趋近直线，栖居其上的蚂蚁永远无法感知其弯曲。但若在双环图形的交叉点放大，则永远看不到直线——蚂蚁会在交叉点察觉自己并非处于欧氏空间，因此双环不是流形。

同理，地球表面是二维流形：任意位置足够放大都会呈现二维平面形态。但由两个圆锥顶点连接构成的双锥面则非流形。

流形解决了数学家的一个难题：形状特性会因其所在空间的性质、维度及嵌入方式而改变。比如将线绳放在桌上连接两端，会形成简单环圈；但若在空中连接两端，借助三维空间让线绳自我穿插，就能打出各种复杂绳结。它们虽代表相同的一维流形（闭合线圈），在二维与三维空间中却呈现不同特性。

为规避这种歧义，数学家聚焦于流形的内在属性。流形的定义特性——任意局部都类似欧氏空间——在此极具价值。由于可将流形任意小块置于欧氏空间中思考，数学家能运用传统微积分技术计算面积体积或描述运动轨迹。

具体操作时，数学家将给定流形划分为若干重叠区块，每个区块用“坐标图”表示——这是一组坐标（数量等于流形维度），用于定位点在流形上的位置。关键在于还需建立描述重叠坐标图间坐标转换的规则。所有这些坐标图的集合称为“图册”。

借助这种图册——将复杂流形的小区域转换为熟悉欧氏空间的工具——人们可以逐块测量探索流形。若要理解函数在流形上的行为，或把握其整体结构，可将问题分解，在不同坐标图的欧氏空间中分别求解，最后整合所有结果得出完整答案。

如今，这种方法已渗透数学与物理学的各个角落。

流形之用
流形对理解宇宙至关重要。爱因斯坦在广义相对论中将时空描述为四维流形，引力则是该流形的曲率表现。我们生活的三维空间本身也是流形——正如所有流形特性，栖居其中的观察者会认为它是欧几里得式的，尽管人类仍在探索其整体形状。

即使在看似不涉及流形的领域，数学家和物理学家也尝试用流形语言重构问题以利用其特性。“物理学的精髓往往在于理解几何，”普林斯顿大学理论物理学家乔纳森·索斯指出，“且常以出人意料的方式呈现。”

以双摆系统为例（即一个摆锤悬挂在另一个摆锤末端）。初始条件的微小变化会导致其运动轨迹天差地别，使得预测和理解其行为异常困难。但若用两个角度（分别描述两段摆臂的位置）表示摆锤构型，则所有可能构型构成的空间形似甜甜圈（环面）——一个流形。环面上每个点代表摆锤的一种可能状态；环面上的路径则对应摆锤在空间中的可能轨迹。这使研究者能将物理问题转化为几何问题，使其更直观易解。同样的方法也被用于研究流体、机器人、量子粒子等系统的运动。

类似地，数学家常将复杂代数方程的解集视为流形以研究其性质；在分析高维数据集（如记录大脑中数千神经元活动的数据）时，他们会考察这些数据点如何分布于低维流形上。

索斯总结道：“追问科学家如何运用流形，如同追问如何运用数字——它们是万物的基石。”

英文来源：

What Is a Manifold?
Introduction
Standing in the middle of a field, we can easily forget that we live on a round planet. We’re so small in comparison to the Earth that from our point of view, it looks flat.
The world is full of such shapes — ones that look flat to an ant living on them, even though they might have a more complicated global structure. Mathematicians call these shapes manifolds. Introduced by Bernhard Riemann in the mid-19th century, manifolds transformed how mathematicians think about space. It was no longer just a physical setting for other mathematical objects, but rather an abstract, well-defined object worth studying in its own right.
This new perspective allowed mathematicians to rigorously explore higher-dimensional spaces — leading to the birth of modern topology, a field dedicated to the study of mathematical spaces like manifolds. Manifolds have also come to occupy a central role in fields such as geometry, dynamical systems, data analysis and physics.
Today, they give mathematicians a common vocabulary for solving all sorts of problems. They’re as fundamental to mathematics as the alphabet is to language. “If I know Cyrillic, do I know Russian?” said Fabrizio Bianchi, a mathematician at the University of Pisa in Italy. “No. But try to learn Russian without learning Cyrillic.”
So what are manifolds, and what kind of vocabulary do they provide?
Ideas Taking Shape
For millennia, geometry meant the study of objects in Euclidean space, the flat space we see around us. “Until the 1800s, ‘space’ meant ‘physical space,’” said José Ferreirós, a philosopher of science at the University of Seville in Spain — the analogue of a line in one dimension, or a flat plane in two dimensions.
In Euclidean space, things behave as expected: The shortest distance between any two points is a straight line. A triangle’s angles add up to 180 degrees. The tools of calculus are reliable and well defined.
But by the early 19th century, some mathematicians had started exploring other kinds of geometric spaces — ones that aren’t flat but rather curved like a sphere or saddle. In these spaces, parallel lines might eventually intersect. A triangle’s angles might add up to more or less than 180 degrees. And doing calculus can become a lot less straightforward.
The mathematical community struggled to accept (or even understand) this shift in geometric thinking.
But some mathematicians wanted to push these ideas even further. One of them was Bernhard Riemann, a shy young man who had originally planned to study theology — his father was a pastor — before being drawn to mathematics. In 1849, he decided to pursue his doctorate under the tutelage of Carl Friedrich Gauss, who had been studying the intrinsic properties of curves and surfaces, independent of the space surrounding them.
In 1854, Riemann was required to deliver a lecture to secure a teaching position at the University of Göttingen. His assigned topic: the foundations of geometry. On June 10, despite a fear of public speaking, he described a new theory in which he generalized Gauss’ ideas about the geometry of surfaces to an arbitrary number of dimensions (and even to infinite dimensions).
Gauss was immediately impressed with the lecture, which involved not just math but also philosophy and physics. But most mathematicians found Riemann’s ideas too vague and abstract to be of much use. “Many scientists and philosophers were saying, ‘This is nonsense,’” Ferreirós said. And so, for decades, the work was largely ignored. Riemann’s lecture didn’t appear in print until 1868, two years after his death.
But by the end of the 19th century, mathematical greats like Henri Poincaré had recognized the importance of Riemann’s ideas. And in 1915, Albert Einstein used them in his general theory of relativity, bringing them out of the realm of philosophical abstraction and into the real world. By the middle of the 20th century, they had become a mathematical staple.
Riemann had introduced a concept that could encompass all possible geometries, in any number of dimensions. A concept that would change how mathematicians view space.
A manifold.
Charted Territory
The term “manifold” comes from Riemann’s Mannigfaltigkeit, which is German for “variety” or “multiplicity.”
A manifold is a space that looks Euclidean when you zoom in on any one of its points. For instance, a circle is a one-dimensional manifold. Zoom in anywhere on it, and it will look like a straight line. An ant living on the circle will never know that it’s actually round. But zoom in on a figure eight, right at the point where it crosses itself, and it will never look like a straight line. The ant will realize at that intersection point that it’s not in a Euclidean space. A figure eight is therefore not a manifold.
Similarly, in two dimensions, the surface of the Earth is a manifold; zoom in far enough anywhere on it, and it’ll look like a flat 2D plane. But the surface of a double cone — a shape consisting of two cones connected at their tips — is not a manifold.
Manifolds address a problem that mathematicians would otherwise have to deal with: A shape’s properties can change depending on the nature and dimension of the space it lives in (and how it sits in that space). For instance, lay a piece of string on a table, and connect its ends without lifting it. You’ll get a simple loop. Now hold the string in the air and tie its ends together. By considering the string in three dimensions, you can pass it over and under itself before you connect the ends, creating all sorts of knots beyond the simple loop. They all represent the same one-dimensional manifold — the looped string — but they have different properties when considered in two versus three dimensions.
Mathematicians avoid such ambiguities by focusing on the manifold’s intrinsic properties. The defining property of manifolds — that at any point, they look Euclidean — is immensely helpful on that front. Because it’s possible to think about any small patch of the manifold in terms of Euclidean space, mathematicians can use traditional calculus techniques to, say, compute its area or volume, or describe movement on it.
To do this, mathematicians divide a given manifold into several overlapping patches and represent each with a “chart” — a set of some number of coordinates (equal to the manifold’s dimension) that tell you where you are on the manifold. Crucially, you also need to write down rules that describe how the coordinates of overlapping charts relate to one another. The collection of all these charts is called an atlas.
You can then use this atlas — whose charts translate smaller regions of your potentially complicated manifold into familiar Euclidean space — to measure and explore the manifold one patch at a time. If you want to understand how a function behaves on a manifold, or get a sense of its global structure, you can break the problem up into pieces, solve each piece on a different chart, in Euclidean space, and then stitch together the results from all the charts in the atlas to get the full answer you’re seeking.
Today, this approach is ubiquitous throughout math and physics.
Manifold Uses
Manifolds are crucial to our understanding of the universe, for one. In his general theory of relativity, Einstein described space-time as a four-dimensional manifold, and gravity as that manifold’s curvature. And the three-dimensional space we see around us is also a manifold — one that, as manifolds do, appears Euclidean to those of us living within it, even though we’re still trying to figure out its global shape.
Even in cases where manifolds don’t seem to be present, mathematicians and physicists try to rewrite their problems in the language of manifolds to make use of their helpful properties. “So much of physics comes down to understanding geometry,” said Jonathan Sorce, a theoretical physicist at Princeton University. “And often in surprising ways.”
Consider a double pendulum, which consists of one pendulum hanging from the end of another. Small changes in the double pendulum’s initial conditions lead it to carve out very different trajectories through space, making its behavior hard to predict and understand. But if you represent the configuration of the pendulum with just two angles (one describing the position of each of its arms), then the space of all possible configurations looks like a doughnut, or torus — a manifold. Each point on this torus represents one possible state of the pendulum; paths on the torus represent the trajectories the pendulum might follow through space. This allows researchers to translate their physical questions about the pendulum into geometric ones, making them more intuitive and easier to solve. This is also how they study the movements of fluids, robots, quantum particles and more.
Similarly, mathematicians often view the solutions to complicated algebraic equations as a manifold to better understand their properties. And they analyze high-dimensional datasets — such as those recording the activity of thousands of neurons in the brain — by looking at how those data points might sit on a lower-dimensional manifold.
Asking how scientists use manifolds is akin to asking how they use numbers, Sorce said. “They are at the foundation of everything.”