具有常长度曲率法向的高秩子流形

中文摘要 AI 辅助

我们研究了秩至少为二的欧几里得子流形的局部几何，这些子流形的曲率法向相对于法丛的平坦部分具有常长度，等价地，其适应的第三基本形式具有常特征值。完整情形已由Di Scala和第三作者完全解决，他们证明了这样的完整子流形具有常主曲率。在正则点附近，我们证明$M$通过平行流形由超曲面$N$生成，$N$的外在因子包含在球面中，并且当维数至少为二时，这些因子具有秩一；反之，这样的因子可以组装起来构造$M$的例子。证明依赖于等参秩定理来排除非脐因子。作为进一步的应用，当曲率法向的内积为常数时，我们证明恰好存在两个特征分布，其中一个是一维且自平行的，这将法丛平坦时任意余维数的曲率齐性结果从Tsukada的结果以及Bryant、Florit和Ziller最近关于超曲面的结果进行了推广。

英文摘要

We study the local geometry of Euclidean submanifolds of rank at least two whose curvature normals with respect to the flat part of the normal bundle have constant length, equivalently, whose adapted third fundamental form has constant eigenvalues. The complete case was fully settled by Di Scala and the third author, who showed that such a complete submanifold has constant principal curvatures. Around a regular point, we show that $M$ is generated, via parallel manifolds, by a hypersurface $N$ whose extrinsic factors are contained in spheres and, when of dimension at least two, have rank one; conversely, such factors can be assembled to construct examples of $M$. The proof relies on the isoparametric rank theorem to rule out non-umbilical factors. As a further application, when the inner products of the curvature normals are constant, we show that there are exactly two eigendistributions, one of which is one-dimensional and autoparallel, extending, for arbitrary codimension when the normal bundle is flat, curvature-homogeneity results of Tsukada and the recent ones of Bryant, Florit, and Ziller for hypersurfaces.

发表机构

• Universidad de Antioquia（安蒂奥基亚大学）
• Universidade Federal de São Carlos（圣卡洛斯联邦大学）
• CIEM-CONICET（CONICET数学与物理研究所）

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