Elasticae in a Riemannian submanifold
Elasticae in a Riemannian submanifold
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黎曼子流形中的弹性体
DOI：
10.18910/3821
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发表时间：
1992
影响因子：
0.4
通讯作者：
N. Koiso
中科院分区：
文献类型：
--
作者：
N. Koiso
对于黎曼流形M中的曲线γ（s），我们定义两个量：长度L（γ）和总平方曲率E（<γ）。一条曲线γ被称为弹性曲线，如果它是泛函E的临界点，且该临界点被限制在具有固定1/2 L0的曲线空间中。弹性的概念是相当古老的。但现代的方法，它在微分几何是相当新的。J. Langer和D.A. Singer在欧几里得空间（[！]）中分类了所有封闭的弹性体，并证明了Palais-Smale条件（C）对黎曼流形中的曲线空间成立（[2]）。本文考虑了限制在子流形中的弹性函数。例如，设M是欧氏空间的一个紧致曲面，C是该曲面中所有给定长度的闭曲线的集合。C中是否有一条闭合曲线使弹性能E最小（定义为欧几里得空间的曲线）？我们将在更一般的情况下肯定地回答这个问题。
For a curve 7(s) in a riemannian manifold M we define two quantities: the length L(γ) and the total square curvature E(<γ). A curve γ is called an elastica if it is a critical point of the functional E restricted to the space of curves of a fixed Inegth L0. The notion of elastica is quite old. But modern approaches to it in differential geometry are rather new. J. Langer and D.A. Singer classified all closed elasticae in the euclidean space ([!]), and showed that Palais-Smale's condition (C) holds for the space of curves in a riemannian manifold ([2]). In this paper we consider elasticae restricted in a submanifold. For example, let M be a compact surface of the euclidean space and C the set of all closed curves of given length in the surface. Is there a closed curve in C which minimizes the elastic energy E (defined as curves of the euclidean space) ? We will affirmatively answer to the question in a more general situation.