Invariants of Lines on Surfaces in $\R^4$

TL;DR

This paper investigates invariants of lines on surfaces in four-dimensional space, characterizing conjugate, asymptotic, and principal tangents, and introduces invariants related to curvature and torsion with applications to rotational surfaces.

Contribution

It introduces new invariants for lines on surfaces in -dimensional space and characterizes key geometric features like conjugate, asymptotic, and principal tangents.

Findings

Invariant of tangent pairs determines conjugacy.

Normal curvature and geodesic torsion characterize asymptotic and principal tangents.

Existence of surfaces with zero invariant k and principal asymptotic lines as helices.

Abstract

Considering the tangent plane at a point to a surface in the four-dimensional Euclidean space, we find an invariant of a pair of two tangents in this plane. If this invariant is zero, the two tangents are said to be conjugate. When the two tangents coincide with a given tangent, then we obtain the normal curvature of this tangent. Asymptotic tangents (curves) are characterized by zero normal curvature. Considering the invariant of the pair of a given tangent and its orthogonal one, we introduce the geodesic torsion of this tangent. We obtain that principal tangents (curves) are characterized by zero geodesic torsion. The invariants $\varkappa$ and $k$ are introduced as the symmetric functions of the two principal normal curvatures. The geometric meaning of the semi-sum $\varkappa$ of the principal normal curvatures is equal (up to a sign) to the curvature of the normal connection of the surface. The number of asymptotic tangents at a point of the surface is determined by the sign of the invariant $k$. In the case $k=0$ there exists a one-parameter family of asymptotic lines, which are principal. We find examples of such surfaces ($k=0$) in the class of the general rotational surfaces (in the sense of Moore). The principal asymptotic lines on these surfaces are helices in the four-dimensional Euclidean space.