Relating second order geometry of manifolds through projections and normal sections
Relating second order geometry of manifolds through projections and normal sections
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通过投影和法线截面关联流形的二阶几何
DOI：
10.5565/publmat6512114
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
R. O. Sinha
中科院分区：
文献类型：
--
作者：
P. B. Riul;R. O. Sinha
我们使用法线截面来关联正则(分别是。$\mathbb{R}^6$(分别.$\mathbb R^5$)和常规(分别为$\mathbb R^5$中的奇异Corank 1)曲面$\mathbb R^4$)。例如，我们展示了如何通过与Steiner的方法不同的一族椭圆族来生成罗马曲面。此外，根据作为正规截面的奇异曲面的曲率轨迹的拓扑类型，给出了奇异3-流形的参数化为2-JET在某一轨道上的必要条件。并通过投影研究了正则情形与奇异情形之间的关系。我们证明了不同类型流形的曲率轨迹之间存在一个投影和法截面的交换图，因此，它们的二阶几何都是相关的。特别地，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$-mathbb R^6$中的正则3-流形和$\mathbb R^4$中的奇异Corank 1曲面的渐近方向联系起来.
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 $\mathbb R^4$.
DOI：
10.1142/9108
发表时间：
2015-12
期刊：
--
影响因子：
--
作者：
S. Izumiya;M. Fuster;M. Ruas;F. Tari
通讯作者：
S. Izumiya;M. Fuster;M. Ruas;F. Tari