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Geodesic distance on the manifold of Riemannian metrics


Speaker: Brian Clarke, Stanford
Location: 2112 MSB
Start time: Mon, Jan 10 2011, 4:10PM

On a fixed closed manifold, I will consider the manifold of all possible Riemannian metrics. This manifold is itself equipped with a canonical Riemannian metric, called the L^2 metric. I will give an explicit formula for the distance, with respect to the L^2 metric, between any two metrics on the base manifold. Additionally, the completion of the manifold of metrics can be described, and I will present an explicit expression for the unique minimal path between any two points in this completion. Time permitting, I will also discuss connections to other areas such as Teichmuller theory and the convergence of Riemannian manifolds.