黎曼在19世纪中叶提出「流形」观念，将「空间」从单纯的物理背景，提升为可被严格研究的抽象对象。时间线上，他1849年在高斯门下攻读博士，1854年6月10日的讲演把曲面几何推广到任意维（甚至无限维）；但这些想法长期被视为过于抽象，直到1868年（他逝世两年后）才刊行。19世纪末（如庞加莱）开始重视，1915年爱因斯坦在广义相对论中采用，并在20世纪中叶成为数学常备工具。

流形的核心定义是「在每一点局部看起来像欧几里得空间」：圆是1维流形（局部像直线），而「8」字形在自交点无法局部像直线，因此不是流形；地球表面是2维流形（局部像平面），但双圆锥的尖端使其失去局部欧式性。为了把局部的「像平直」转化为可计算的结构，数学家用多个重叠小区域建立坐标「图」，每张图用与流形维度相同数目的坐标，并给出重叠区的换算规则；所有图组成「图册」，使得可在每个局部套用传统微积分，再把结果拼接成全局结论。

在应用上，广义相对论把时空建模为4维流形，重力对应其曲率；我们日常感知的空间则可视为3维流形。动力系统中，双摆可用两个角度描述其构型空间，该空间拓扑上像甜甜圈（环面）这个流形：每个点代表一种状态、曲线代表演化轨迹，将「难预测的运动」转成「几何上的路径」。同样思路也用于代数方程解集、以及把包含上千个神经元活动等高维资料，理解为落在较低维流形上的结构。

Riemann’s mid-19th-century idea of a manifold reframed “space” as an abstract object, not just a physical stage. The timeline is explicit: in 1849 he pursued a doctorate under Gauss; on June 10, 1854 he lectured at Göttingen, extending surface geometry to arbitrary (even infinite) dimensions; the work was printed only in 1868, two years after his death. It gained traction by the late 19th century, entered physics decisively in 1915 via Einstein’s general relativity, and became standard by the mid-20th century.

A manifold is defined by a local numerical criterion: at every point it looks Euclidean. A circle is a 1-dimensional manifold because small neighborhoods resemble a line; a figure eight fails at its self-intersection. In 2 dimensions, Earth’s surface is locally a plane, while a double cone fails at the tip. To compute intrinsically, mathematicians cover a manifold with overlapping coordinate “charts,” each using as many coordinates as the manifold’s dimension, and specify transition rules on overlaps; the full collection is an “atlas,” enabling calculus locally and global results by stitching.

Manifolds supply a common language across fields: spacetime in relativity is a 4-dimensional manifold whose curvature models gravity, and ordinary space can be treated as a 3-dimensional manifold. In dynamics, a double pendulum’s configurations can be encoded by 2 angles, making the state space a torus manifold; points are states and paths are trajectories, turning sensitivity to initial conditions into geometry. Similar manifold viewpoints organize solution sets of algebraic equations and interpret high-dimensional data—e.g., recordings from thousands of neurons—as lying on lower-dimensional structure.