通过投影和法截面联系流形的二阶几何
Relating second order geometry of manifolds through projections and normal sections
摘要：
我们用法截面来描述正则（分别是）的曲率轨迹。奇异余秩1）$\mathbb{R}^6$中的3-流形（分别为$\mathbb R^5$）与常规（分别奇异corank 1）表面在$\mathbb R^5$（分别）。$\mathbb R^4$）。例如，我们将展示如何生成一个罗马曲面的椭圆不同的施泰纳的方式家庭。进一步，我们根据作为法截面得到的奇异曲面的曲率轨迹的拓扑型，给出了奇异3-流形的参数化的2-喷流在某个轨道上的必要条件.我们还通过投影研究了正则情形和奇异情形之间的关系。我们表明，有一个交换图表的投影和正常的部分，涉及的曲率轨迹的不同类型的流形，因此，二阶几何的所有这些都是相关的。特别是我们定义了$\mathbb R^5$中奇异corank 1 3-流形的渐近方向，并将它们与$\mathbb R^6$中正则3-流形和$\mathbb R^4$中奇异corank 1曲面的渐近方向联系起来。
英文摘要：
We use normal sections to relate the curvature locus of regular (resp. singular corank 1) 3-manifolds in $\mathbb{R}^6$ (resp. $\mathbb R^5$) with regular (resp. singular corank 1) surfaces in $\mathbb R^5$ (resp. $\mathbb R^4$). For example we show how to generate a Roman surface by a family of ellipses different to Steiner's way. Furthermore, we give necessary conditions for the 2-jet of the parametrisation of a singular 3-manifold to be in a certain orbit in terms of the topological types of the curvature loci of the singular surfaces obtained as normal sections. We also study the relations between the regular and singular cases through projections. We show there is a commutative diagram of projections and normal sections which relates the curvature loci of the different types of manifolds, and therefore, that the second order geometry of all of them is related. In particular we define asymptotic directions for singular corank 1 3-manifolds in $\mathbb R^5$ and relate them to asymptotic directions of regular 3-manifolds in $\mathbb R^6$ and singular corank 1 surfaces in $\mathbb R^4$.
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