The Gauss Section of a Riemannian Immersion
The Gauss Section of a Riemannian Immersion
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黎曼浸没的高斯部分
DOI：
10.1112/jlms/s2-33.1.157
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发表时间：
1986
影响因子：
1.2
通讯作者：
C. Wood
中科院分区：
文献类型：
--
作者：
C. Wood
黎曼流形[3]在黎曼纤维丛截面的上下文中。在§ 1中，我们引入了垂直能量泛函并计算了它的第一变分（定理1）。与调和映射的情况相反，出现了挠率张量，这可以看作是垂直能量定义中隐含的不对称性的结果。因此，为了得到精确的欧拉-拉格朗日方程（定理2），只需将注意力限制在那些具有相应不对称性的变化--垂直变化上。我们选择了调和截面这个短语来描述这样的临界点，尽管一般来说这些都不是调和映射。在§ 2中，我们的目标是将Obata [6]和Ruh和Vilms [7]的结果结合起来，他们研究了欧氏空间（或者更一般地说[6]，常曲空间）的等距浸入子流形。这样的浸入产生到适当的格拉斯曼流形的高斯映射，并且高斯映射的共形性和调和性（分别在[6，7]中）被示出为对应于浸入曲率的条件。为了消除周围空间中常曲率的限制，我们考虑高斯提升到适当的格拉斯曼丛。我们导出的垂直共形性和垂直调和性的条件（定理4-7）可以分别看作是高斯方程和科达齐方程的表现。等价地，它们可以被看作是某个Weitzenbock公式的切向分量和法向分量，其中显然最不具有几何意义的项（所谓的粗糙拉普拉斯算子）提供了浸入与其高斯升力之间的联系。希望这种方法可以帮助揭示其他类型的黎曼（/-结构，其例子出现在[10，11]中。我想借此机会感谢詹姆斯·埃尔斯教授提出的这些以及许多其他想法，作为博士学位的可能课题。论文[9]。关于最近关于共形浸没曲面的高斯升力的工作的细节，我们请感兴趣的读者参阅他的综述文章[2]。
Riemannian manifolds [3] in the context of sections of Riemannian fibre bundles. This is then used to investigate the geometry of an isometrically immersed Riemannian submanifold.In § 1 we introduce the vertical energy functional and calculate its first variation (Theorem 1). In contrast to the case of harmonic maps is the appearance of a torsion tensor, which may be seen as a consequence of the asymmetry implicit in the definition of vertical energy. To obtain nice Euler-Lagrange equations (Theorem 2), it therefore suffices to restrict attention to those variations with the corresponding asymmetry—the vertical variations. We have chosen the phrase harmonic section to describe such critical points, although in general these will not be harmonic maps. In § 2 our aim is to bring together results of Obata [6] and Ruh and Vilms [7], who studied isometrically immersed submanifolds of Euclidean space (or, more generally [6], constantly-curved space). Such immersions give rise to a Gauss map into an appropriate Grassmann manifold, and the conformality and harmonicity of the Gauss map were shown (in [6, 7], respectively) to correspond to conditions on the curvature of the immersion. In order to remove the limitation of constant curvature in the ambient space, we consider instead the Gauss lift to an appropriate Grassmann bundle. The conditions we derive for its vertical conformality and vertical harmonicity (Theorems 4-7) may be seen as manifestations of the equations of Gauss and Codazzi, respectively. Equivalently, they rhay be viewed as the tangential and normal components of a certain Weitzenbock formula, in which the term of apparently least geometrical significance (the so-called rough Laplacian) provides the link between the immersion and its Gauss lift. It is hoped that this approach may help to throw light on other types of Riemannian (/-structures, examples of which appear in [10, 11]. I should like to take this opportunity to thank Professor James Eells for suggesting these, and many other ideas, as possible topics for inclusion in a Ph. D. Thesis [9]. For details of recent work concerning the Gauss lift of a conformally immersed surface, we refer the interested reader to his survey article [2].