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AG Stochastische Geometrie am 15.5.2009

Gil Solanes: Total curvature in hyperbolic space

We present a Gauss-Bonnet formula for the integral of the
extrinsic curvature of complete surfaces in hyperbolic space. This
formula contains two terms, both of them described in the language of
integral geometry.

The first term is related to the the conformal (Möbius) geometry of the
ideal boundary. It can be described as the (Möbius invariant) measure of
point pairs in "non-trivial" position with respect to the curve.
Moreover, it can be represented as bilocal double integral on the curve
at infinity, similar to those appearing in (conformally invariant) knot
energies.

The second term is related to the hyperbolic geometry of the surface,
and can be interpreted as a truncated area. More precisely, it is given
by the measure of geodesic lines intersecting the surface twice. This
notion is motivated by Banchoff-Pohl's definition of the area enclosed
by space curves.

Andreas Bernig: Integral geometry of transitive group actions

I will give a survey of recent results on the integral geometry
of groups
which act transitively on the unit sphere of some finite-dimensional vector
space. There are 6 infinite lists of such groups and 3 exceptional groups.
For each of these groups, one can in principle write down a basis of the
vector space of invariant valuations and a version of the principal kinematic
formula.