Discussing Knarr's 2-Surface of $\mathbb{R}$ ⁴ which Generates the First Single Shift Plane

Abstract

The generating line of the first single shift plane (cf. [11, p. 435]) is a 2-surface of $\mathbb{R}$ ⁴ which we call the the affine part $\Lambda _{Kn}^a$ of Knarr's surface. We compute all affinities leaving $\Lambda _{Kn}^a$ invariant. After embedding $\mathbb{R}$ ⁴ into PG(4,$\mathbb{R}$) we calculate the uniquely determined projective closure Λ _Kn of $\Lambda _{Kn}^a$. Using a suitable projection we transform questions on Knarr's surface to questions on Cayley's surface in PG(3,$\mathbb{R}$). In this way we determine all planes carrying 1-dimensional algebraic varieties of Λ _Kn . We exhibit all automorphic collineations of Λ _Kn .