

Iskovskikh Seminar
May 21, 2015 13:30, Moscow, Steklov Mathematical Institute, room 540






Clifford and Euclidean translations of circles
Niels Lubbes^{} 
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Abstract:
We consider surfaces in the threesphere that admit at least two
circles through a generic closed point. Such surfaces are called
"celestials" and can be obtained by moving a circle along a closed
loop in at least two different ways. The radius of a circle is in
general allowed to change during its motion through the threesphere,
however in this talk we consider the case where the radius remains
fixed. Moreover, the motions we consider are translations. A
translation is an isometry where every point moves with the same
distance. We consider the projective threesphere as a CayleyKlein
model for both elliptic geometry (aka Clifford geometry) and Euclidean
geometry. For example the Clifford torus is wellknown to be the
Clifford translation of a great circle along a great circle. The goal
of this presentation is to answer the following question:
Can a celestial in the threesphere be—up to Moebius
equivalence—both the Clifford and Euclidean translation of a
circle?
For this purpose we propose elliptic and Euclidean invariants for
surfaces in the spherical models. If time permits we give an overview
of recent characterizations of celestials in terms of these
invariants.
Language: English

