A differentiable manifold is a topological space that is locally similar to Euclidean space, allowing calculus to be applied. Each point has a neighborhood that is homeomorphic to an open subset of [乳胶]\mathbb{R}^n[/latex]. These local coordinate systems, called charts, are related by smooth transition functions, forming an atlas that defines the manifold’s differentiable structure.
可微流形（geom）
1854
- Bernhard Riemann
A differentiable manifold is the central object of study in differential geometry. The concept formalizes the idea of a “curved space” of any dimension. While globally a manifold can be complex (like a sphere or a torus), locally, around any point, it looks like a flat piece of Euclidean space. This local “flatness” is key, as it allows us to use the tools of multivariable calculus.
The formal definition involves a set of points M, a topology on M, and an atlas. An atlas is a collection of charts, where each chart is a pair (U, φ), with U being an open subset of M and φ being a homeomorphism from U to an open subset of [latex]\mathbb{R}^n[/latex]. For any two overlapping charts, (U, φ) and (V, ψ), the transition map [latex]\psi \circ \phi^{-1}[/latex] from [latex]\phi(U \cap V)[/latex] to [latex]\psi(U \cap V)[/latex] must be a diffeomorphism (infinitely differentiable with a differentiable inverse). This compatibility condition ensures that calculus performed in one coordinate system is consistent with calculus performed in another.
这种结构允许在流形上定义切空间、矢量场和微分形式，而不受任何特定坐标系的限制。它提供了一个从本质上研究几何的框架，而无需将空间嵌入到更高维度的环境空间中。
UNESCO Nomenclature: 1204
- 几何学
类型
抽象系统
中断
基础
使用方法
广泛使用
前体
- 欧几里得几何
- 非欧几里得几何（罗巴切夫斯基、博雅伊）
- 卡尔·弗里德里希·高斯的曲面理论
- 勒内·笛卡尔的坐标系
- 拓扑学的早期概念
应用
- 广义相对论（时空被建模为 4d 洛伦兹流形）
- 机器人技术（机器人的配置空间是流形）
- 计算机图形学（表示复杂表面）
- 弦理论
- 经典力学（相空间是辛流形）
专利：
NA
潜在的创新想法
Related to: manifold, topology, differentiable structure, atlas, chart, euclidean space, calculus, geometry.
迎接新挑战
机械工程师、项目、工艺工程师或研发经理
可在短时间内接受新的挑战。
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历史背景
(如果日期不详或不相关，例如 "流体力学"，则对其显著出现的时间作了四舍五入的估计）。
相关发明、创新和技术原理
