Geometry of surfaces in \mathbb R⁵ through projections and normal sections

2020/10/21 by Silva, Jorge Deolindo, Sinha, Raúl Oset
#53A05 #57R45 #58K05 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2010.10976

Abstract

We study the geometry of surfaces in \mathbb R⁵ by relating it to the geometry of regular and singular surfaces in \mathbb R⁴ obtained by orthogonal projections. In particular, we obtain relations between asymptotic directions, which are not second order geometry for surfaces in \mathbb R⁵ but are in \mathbb R⁴. 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 R⁶ and again relate asymptotic directions and contact with spheres by defining an appropriate umbilic curvature for 3-manifolds.

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