A Symplectic Look at Surfaces of Revolution
A Symplectic Look at Surfaces of Revolution
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旋转表面的辛观察
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通讯作者：
Andrew D. Hwang
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作者：
Andrew D. Hwang
非正式地，旋转曲面是一个2维黎曼流形，它具有等距圆作用。旋转曲面是微分几何中最简单的对象之一;度规由一个真实的变量的单一函数确定，因此可以通过求解常微分方程来指定。一个函数x：n → R是一个“轨道参数”，如果x的每个水平集都是一个轨道。给定一个轨道参数，轨道的“轮廓”是一个决定轨道长度的函数。例如，当一个函数的图形绕R中的一个轴旋转时，“明显的”轨道参数是沿旋转轴的笛卡尔坐标x沿着，而函数本身是一个轮廓。本文从一个基本的内禀轨道参数和轮廓函数出发构造旋转曲面。出发点是阿基米德定理，其证明是当今一个简单的微积分练习。设S R为单位球面，视其为固定任意直径的旋转曲面。S的“区域”是由垂直于直径的两个平面界定的子集，并且区域的“高度”是其边界平面之间的距离。
Informally, a surface of revolution is a 2-dimensional Riemannian manifold Σ equipped with an isometric circle action. Surfaces of revolution are among the simplest objects in differential geometry; the metric is determined by a single function of one real variable, hence can be specified by solving an ordinary differential equation. A function x : Σ → R is an “orbit parameter” if each level set of x is a single orbit. Given an orbit parameter, a “profile” for Σ is a function that determines the lengths of the orbits. For example, when the graph of a function ξ is revolved about an axis in R, the “obvious” orbit parameter is a Cartesian coordinate x along the axis of revolution, and ξ itself is a profile. This note constructs surfaces of revolution from an elementary, intrinsic orbit parameter and profile function. The point of departure is a theorem of Archimedes, whose proof is nowadays an easy calculus exercise. Let S ⊂ R be the unit sphere, regarded as a surface of revolution by fixing an arbitrary diameter. A “zone” of S is a subset bounded by two planes perpendicular to the diameter, and the “height” of a zone is the distance between its bounding planes.