Ribaucour Transformations Revisited
Ribaucour Transformations Revisited
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重温里鲍库尔的转变
DOI：
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发表时间：
2004
期刊：
影响因子：
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通讯作者：
K. Tenenblat
中科院分区：
文献类型：
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作者：
A. Corro;K. Tenenblat
我们提出了空间形式子流形的里鲍库变换的修订定义，具有平坦的法束，受到经典定义和最近扩展的启发。本文引入的定义提供了对保留曲率线的此类变换的几何方面的精确处理，并且它可以应用于主曲率重数大于 1 的子流形。我们用微分方程来描述这种变换，并研究它的一些性质。我们证明，n 维球体或超平面可以通过 Ribaucour 变换局部关联到 R 的任何给定超曲面 M，其中允许 n 个正交主方向向量场。作为 Ribaucour 变换的应用，我们将主曲率恒定重数为 1 的 Dupin 超曲面表征为由 Ribaucour 变换关联的 (n−1) 维 Dupin 子流形组成的流形。
We present a revised definition of a Ribaucour transformation for submanifolds of space forms, with flat normal bundle, motivated by the classical definition and by more recent extensions. The definition introduced in this paper, provides a precise treatment of the geometric aspect of such transformations preserving lines of curvature and it can be applied to submanifolds whose principal curvatures have multiplicities bigger than one. We characterize this transformation in terms of differential equations and we study some of its properties. We show that an n-dimensional sphere or hyperplane can be locally associated by a Ribaucour transformation to any given hypersurface M of R, which admits n orthogonal principal direction vector fields. As an application of Ribaucour transformations, we characterize the Dupin hypersurfaces which have a principal curvature of constant multiplicity one, as a manifold foliated by (n−1)-dimensional Dupin submanifolds associated by Ribaucour transformations.