Germain曲率：在Sophie Germain之后求曲面的中曲率
Germain Curvature: The Case for Naming the Mean Curvature of a Surface after Sophie Germain
摘要：
我们如何描述曲面的形状？现在大家都知道，一个表面的形状是通过测量它在每一点上的弯曲程度来确定的。从这些测量中，可以识别最大和最小曲率的方向，即主曲率，并通过取主曲率的平均值和乘积来构造两个表面不变量。主曲率的乘积描述了曲面的内禀曲率，在微分几何中有着深远的重要性-由高斯定理和高斯-博内定理证明。这种曲率通常被称为高斯曲率后，卡尔弗里德里希高斯，他的重大贡献，新兴领域的微分几何在他的1828年的工作。平均值或中曲率是表面形状的外在度量-也就是说，形状必须嵌入到要测量的高维空间中。从1811年开始，数学家索菲·热尔曼（Sophie Germain）在1821年和1826年的努力中达到顶峰，她确定中曲率是描述振动板形状的适当度量。她的假设直接导致了描述弹性薄板行为的方程。在给高斯的信中，她描述了她的“中曲率球“的概念，可以在表面上的每一点上识别。这一贡献刺激了一段时间的快速发展，弹性和几何，但热尔曼尚未收到应有的信贷推导的数学和物理意义的中曲率。从原始证据中可以清楚地看出，这种中曲率的测量应该以Sophie Germain的名字命名。
英文摘要：
How do we characterize the shape of a surface? It is now well understood that the shape of a surface is determined by measuring how curved it is at each point. From these measurements, one can identify the directions of largest and smallest curvature, i.e. the principal curvatures, and construct two surface invariants by taking the average and the product of the principal curvatures. The product of the principal curvatures describes the intrinsic curvature of a surface, and has profound importance in differential geometry - evidenced by the Gauss's Theorema Egregium and the Gauss-Bonnet theorem. This curvature is commonly referred to as the Gaussian Curvature after Carl Friedrich Gauss, following his significant contributions to the emerging field of differential geometry in his 1828 work. The average, or mean curvature, is an extrinsic measure of the shape of a surface - that is, the shape must be embedded in a higher dimensional space to be measured. Beginning in 1811, and culminating in efforts in 1821 and 1826, the mathematician Sophie Germain identified the mean curvature as the appropriate measure for describing the shape of vibrating plates. Her hypothesis leads directly to the equations describing the behavior of thin, elastic plates. In letters to Gauss, she described her notion of a "sphere of mean curvature" that can be identified at each point on the surface. This contribution stimulated a period of rapid development in elasticity and geometry, and yet Germain has not yet received due credit for deducing the mathematical and physical significance of mean curvature. It is clear from the primary source evidence that this measure of mean curvature should bear the name of Sophie Germain.
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