- 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, 13th Jul 2010
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 equal-mass 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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