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Geometry of surfaces in $$\mathbb R^5$$ R 5 through projections and normal sections
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas ( IF 1.8 ) Pub Date : 2021-03-20 , DOI: 10.1007/s13398-021-01019-1
J. L. Deolindo-Silva , R. Oset Sinha

We study the geometry of surfaces in \(\mathbb {R}^5\) by relating it to the geometry of regular and singular surfaces in \(\mathbb {R}^4\) obtained by orthogonal projections. In particular, we obtain relations between asymptotic directions, which are not second order geometry for surfaces in \(\mathbb {R}^5\) but are in \(\mathbb {R}^4\). We also relate the umbilic curvatures of each type of surface and their contact with spheres. We then consider the surfaces as normal sections of 3-manifolds in \(\mathbb {R}^6\) and again relate asymptotic directions and contact with spheres by defining an appropriate umbilic curvature for 3-manifolds.



中文翻译:

$$ \ mathbb R ^ 5 $$ R 5中的曲面的几何形状(通过投影和法线截面)

我们通过将\(\ mathbb {R} ^ 5 \)中的曲面的几何与通过正交投影获得的\(\ mathbb {R} ^ 4 \)中的规则曲面和奇异曲面的几何相关联来研究该曲面的几何。特别是,我们获得渐近方向之间的关系,它们不是\(\ mathbb {R} ^ 5 \)中的曲面的二阶几何,而是在\(\ mathbb {R} ^ 4 \)中的曲面的关系。我们还关联了每种类型的曲面的脐曲率及其与球体的接触。然后,我们将曲面视为\(\ mathbb {R} ^ 6 \)中的3个歧管的法线截面,并通过为3个歧管定义适当的脐曲率,再次关联渐近方向和与球体的接触。

更新日期:2021-03-21
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