Local geometry of surfaces in $\mathbf R^4$

TL;DR

This paper explores the local geometric properties of surfaces in four-dimensional space, focusing on curvature-related concepts like the indicatrix, characteristic curve, and classifications of points based on curvature.

Contribution

It introduces the curvature ellipse and characteristic curve for surfaces in R^4, and discusses their projective duality and point classifications, providing new insights into surface geometry in higher dimensions.

Findings

Defined the indicatrix and curvature ellipse in R^4

Established the projective duality between key geometric objects

Classified points on surfaces as elliptic, parabolic, hyperbolic, and inflection points

Abstract

The indicatrix or curvature ellipse and the characteristic curve of a surface in $\mathbf R^4$ are presented, as well as the projective duality connecting them. The characterisation of points in the surfaces as elliptic, parabolic and hyperbolic points, and the inflection points, are also discussed.