Frank Morgan

The celebrated Gauss-Bonnet formula has a nice generalization to surfaces with densities discovered by my 2004 undergraduate research Geometry Group. The classical Gauss-Bonnet formula relates the integral of the Gauss curvature G over a disc D to the integral over its boundary of the geodesic curvature :

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One can weight arclength and area by densities:

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Surfaces with density appear throughout mathematics, including probability theory and Perelman’s recent proof of the Poincaré Conjecture (see Chapter 18 of the 2009 edition of my Geometric Measure Theory book). Important examples include quotients of Riemannian manifolds by symmetries and Gauss space, defined as Rⁿ with Gaussian density  c exp(-r²).

The generalized Gauss curvature is given by

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Ivan Corwin and I have just written an article about this. The formula for how Gauss curvature changes under a conformal change of metric is a simple special case.