Geometry of surfaces in $\mathbb R^5$ through projections and normal sections

TL;DR

This paper explores the geometry of surfaces in five-dimensional space by analyzing their projections and sections, establishing relationships between asymptotic directions, umbilic curvatures, and contact with spheres across different dimensions.

Contribution

It introduces new relations between surface geometries in  and , including asymptotic directions and umbilic curvatures, through projections and normal sections.

Findings

Relations between asymptotic directions in  and  surfaces.

Connections between umbilic curvatures and sphere contact.

Framework for analyzing 3-manifolds via surface projections.

Abstract

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.