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

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.