# Gruppe for geometri/topologi ved Institutt for matematiske fag, NTNU

## Seminar våren 2010

Lars Sydnes: Möbius Transformations and the Study of Celestial Bodies
21. januar 2010, rom 734, Sentralbygg 2, 14.15 - 16.00
Abstract: The set of similarity classes of (not totally degenerate) triangles can in a natural way be identified with the sphere.
In the kinematic geometry of the three body problem, this sphere comes with a riemannian metric with a fixed constant curvature. However, when we demand this fixed constant curvature, and change the mass distribution, we have to modify the mapping from triangles to points on this sphere. We will see that this modification can be described by a Möbius transformation keeping the equator fixed. (This will correspond to an isometry of the hyperbolic plane.)
During the talk, I hope to give a proof of this claim. The bacground material for this proof consists of discussion of (i) the conformal structure of the sphere, (ii) equivariance of the Hopf map with respect to some important group actions, and (iii) the construction of Jacobi vectors depending on the mass distribution for the three body problem.