SC: Surfaces containing two circles through a general point in higher dimensions
Speaker: Niels Lubbes (RICAM)
Date: June 9, 2017 10:15
Location: SP2 416-1
We show new examples of surfaces that contain many circles and have many angle-preserving symmetries. It follows from Liouville's theorem that angle-preserving transformations in n-space are Moebius transformations. In Moebius geometry we consider the projectivization of the n-dimensional unit sphere: $S^n$. Problem. Classify surfaces in $S^n$ that contain two circles through a smooth point and are $G$-orbits, where $G$ is any subgroup of the Moebius transformations. As a first step we may assume that $G$ is a subgroup of $SE(3)$, since Euclidean isometries are Moebius transformations. It is classically known that a 2-dimensional $G$-orbit in Euclidean 3-space is either a plane, a sphere or a circular cylinder. The corresponding orbits in $S^3$ are---via the stereographic projection---either $S^2$ or the spindle cyclide. If we consider $G$ to be a subgroup of the Euclidean similarities of 3-space then---aside the surfaces whose 2-dimensional automorphism group are isometries---the circular cone is also a 2-dimensional $G$-orbit in Euclidean 3-space. Its inverse stereographic projection into $S^3$ is known as the horn cyclide. What if $dim(G)2$? In this case, $X$ is $S^2$ or, surprisingly, the Veronese surface in $S^4$ [Kollár, 2016]. Now what if $dim(G)=2$? One example is the ring cyclide in $S^3$. Up to now, the $G$-orbits lead to classical and elementary surfaces. In the talk we will discuss the classification of such surfaces in higher dimensions up to Moebius equivalence.