RICAM Events

Group Seminar

SC: Surfaces containing two circles through a general point in higher dimensions
Speaker: Niels Lubbes
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.

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