Sol and the AGM

Sol and the AGM

Richard Schwartz, Brown University and Institute for Advanced Study

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Sol is one of the 8 Thurston geometries, a 3-dimensional solvable Lie group equipped with its left-invariant metric.  The Riemannian geometry of Sol is pretty strange because it has sectional curvatures of both signs.  Our result is an exact criterion for a geodesic segment in Sol to be a distance minimizer. The criterion is given as an inequality involving the arithmetic-geometric mean of Gauss and is related to the periods of a classic Hamiltonian flow on the 2-sphere. 

I will also explain why this characterization implies that the metric spheres in Sol are topological spheres. I'll demonstrate the main points of the proofs with some computer animations.

This is joint work with Matei Coiculescu.