#### Group Seminar

##### 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.

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