Tight Immersions and Total Absolute Curvature
Tight Immersions and Total Absolute Curvature
复制标题
紧密浸没和总绝对曲率
DOI：
10.1112/blms/3.2.129
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发表时间：
1971
影响因子：
0.9
通讯作者：
T. Willmore
中科院分区：
文献类型：
--
作者：
T. Willmore
浸入流形的全绝对曲率理论是数学中称为整体微分几何的分支的一部分。一个紧可微流形浸入另一个流形，通常是欧几里得空间，这就产生了各种曲率测度。通过积分，可以将浸入与一个真实的数、浸入的总绝对曲率相关联。我们现在考虑这些数的下确界在整个浸入类上，并且所得的数仅取决于给定的流形。这样我们就得到了流形的不变量，这些不变量依赖于它的拓扑结构，也许也依赖于它的可微结构。达到总绝对曲率下确界的浸入称为”紧”浸入。一个自然的问题是考虑什么流形紧浸入是可能的，什么是图像的几何属性和浸入映射的性质。全绝对曲率理论可以认为起源于1929年W。Fenchel [15].在那篇论文中，证明了，对于可微类^3的E3的闭空间曲线C，<\ds^ 2，其中K是普通曲率。此外，等式发生当且仅当C是平面凸曲线。
The theory of total absolute curvature of immersed manifolds is part of the branch of mathematics known as global differential geometry. A compact differentiable manifold is immersed in another manifold, usually euclidean space, and this gives rise to various curvature measures. By integration, it is possible to associate with the immersion a real number, the total absolute curvature of the immersion. We now consider the infimum of these numbers taken over the whole class of immersions, and the resulting number depends only on the given manifold. In this way we obtain invariants of the manifold which depend on its topological structure and perhaps on its differentiable structure too. An immersion for which the total absolute curvature infimum is attained is called" tight". A natural problem is to consider for what manifolds tight immersions are possible, and what are the geometrical properties of the image and the nature of the immersion mapping. The theory of total absolute curvature can be considered to have originated in 1929 with a paper of W. Fenchel [15]. In that paper it was proved that, for closed space-curves C of E3 of differentiability class^ 3,<\ds^ 2, where K is the ordinary curvature. Moreover equality takes place if and only if C is a plane convex curve.