 CARMA CAT(0) SEMINAR
 Speaker: Dr Ian Benn, CARMA, The University of Newcastle
 Title: The Riemannian Centre of Mass
 Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle
 Time and Date: 1:00 pm, Tue, 13^{th} Jul 2010
 Abstract:
In Euclidean space the medians of a triangle meet at a point that divides each median in the ratio 2 to 1. That point is called the centroid. Cinderella tells us that the medians of a triangle in hyperbolic space meet at a point, but the medians do not divide each other in any ﬁxed ratio. What characterises that point? One answer is that it is the centre of mass of equalmass particles placed at the vertices. I will outline how one can deﬁne the centre of mass of a set of particles (points) in a Riemannina manifold, and how one can understand this in terms of the exponential map. This centre of mass, or geometric mean, is sometimes called the Karcher mean (apparently ﬁrst introduced by Cartan!). I will attempt to show what this tells us about the medians of a triangle.
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