Vol. 4, No. 2, 2011

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The Gauss–Bonnet formula on surfaces with densities

Ivan Corwin and Frank Morgan

Vol. 4 (2011), No. 2, 199–202

The celebrated Gauss–Bonnet formula has a nice generalization to surfaces with densities, in which both arclength and area are weighted by positive functions. Surfaces with densities, especially when arclength and area are weighted by the same factor, appear throughout mathematics, including probability theory and Perelman’s recent proof of the Poincaré conjecture.

Gauss–Bonnet, density
Mathematical Subject Classification 2010
Primary: 53B20
Received: 30 June 2011
Revised: 11 July 2011
Accepted: 11 July 2011
Published: 17 January 2012

Communicated by Michael Dorff
Ivan Corwin
Courant Institute of Mathematics
New York University
251 Mercer Street
New York, NY 10012
United States
Frank Morgan
Department of Mathematics and Statistics
Williams College
18 Hoxsey Street
Williamstown, MA 01267
United States