Riemannian Manifold

A Riemannian manifold is locally an Euclidean space. An embedded surface (e.g. the Earth) is an example. It defines a Riemannian tensors field (first fundamental form) in parametric space which encode length deformations. https://en.wikipedia.org/wiki/Parametric_surface#First_fundamental_form.

A parametric surface is a surface in the Euclidean space R3 {\displaystyle \mathbb {R} ^{3}} which is defined by a parametric equation with two parameters r:R2→R3 {\displaystyle \mathbf {r} :\mathbb {R} ^{2}\to \mathbb {R} ^{3}}. Parametric representation is a very general way to specify a surface, as well as implicit representation. Surfaces that occur in two of the main theorems of vector calculus, Stokes’ theorem and the divergence theorem, are frequently given in a parametric form. The curvature and arc length of curves on the surface, surface area, differential geometric invariants such as the first and second fundamental forms, Gaussian, mean, and principal curvatures can all be computed from a given parametrization.

黎曼流形局部是欧几里得空间。嵌入式表面(例如地球)就是一个例子。它在参数空间中定义了编码长度变形的黎曼张量场(第一种基本形式)。https://en.wikipedia.org/wiki/parametric_surface#first_fundamental_form 参数化表面是欧氏空间R3 {\displaystyle \mathbb {r} ^{3}} 中定义的表面，具有两个参数的参数方程 r:r2→r3 {\displaystyle \mathbf {r} :\mathbb {r} ^{2}\to \mathbb {r} ^{3}}。参数表示是一种非常通用的方式来指定曲面以及隐式表示。出现在向量微积分的两个主要定理中的表面，斯托克斯定理和发散定理，通常以参数形式给出。曲率和弧长在表面、表面积、差分几何不变量(如第一和第二基本形式、高斯、平均值和主曲率)中的曲率和弧长都可以从给定的参数化中计算出来。

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