{"id":37,"date":"2009-10-17T08:43:31","date_gmt":"2009-10-17T12:43:31","guid":{"rendered":"http:\/\/blogs.williams.edu\/Morgan\/?p=37"},"modified":"2011-06-15T10:07:38","modified_gmt":"2011-06-15T15:07:38","slug":"gauss-bonnet-with-densities","status":"publish","type":"post","link":"https:\/\/sites.williams.edu\/Morgan\/2009\/10\/17\/gauss-bonnet-with-densities\/","title":{"rendered":"Gauss-Bonnet with Densities"},"content":{"rendered":"<p>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 <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Ckappa&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\\kappa' title='\\kappa' class='latex' \/>:<\/p>\n<p style=\"text-align: center\"><img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cint_%7B%5Cpartial+D%7D%5Ckappa+%2B+%5Cint_DG+%3D+2%5Cpi&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\\int_{\\partial D}\\kappa + \\int_DG = 2\\pi' title='\\int_{\\partial D}\\kappa + \\int_DG = 2\\pi' class='latex' \/>.<\/p>\n<p style=\"text-align: left\">One can weight arclength and area by densities:<\/p>\n<p align=\"center\"><img src='https:\/\/s0.wp.com\/latex.php?latex=ds+%3D+%5Cdelta_1+ds_0%2C+dA+%3D+%5Cdelta_2+dA_0&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='ds = \\delta_1 ds_0, dA = \\delta_2 dA_0' title='ds = \\delta_1 ds_0, dA = \\delta_2 dA_0' class='latex' \/>.<\/p>\n<p style=\"text-align: left\">Surfaces with density appear throughout mathematics, including probability theory and Perelman\u2019s recent proof of the Poincar\u00e9 Conjecture (see Chapter 18 of the 2009 edition of my <a href=\"http:\/\/www.elsevierdirect.com:80\/product.jsp?isbn=9780123744449\">Geometric Measure Theory book<\/a>). Important examples include quotients of Riemannian manifolds by symmetries and Gauss space, defined as <strong>R<\/strong><sup>n<\/sup> with Gaussian density\u00a0 c exp(-r<sup>2<\/sup>).<\/p>\n<p>The generalized Gauss curvature is given by<\/p>\n<p align=\"center\"><em><img src='https:\/\/s0.wp.com\/latex.php?latex=G%5E%5Cprime+%3D+G-%5CDelta+log+%5Cdelta_1&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='G^\\prime = G-\\Delta log \\delta_1' title='G^\\prime = G-\\Delta log \\delta_1' class='latex' \/>.<\/em><\/p>\n<p style=\"text-align: left\"><a href=\"http:\/\/www.cims.nyu.edu\/~corwin\/\">Ivan Corwin<\/a> and I have just written <a href=\"http:\/\/www.williams.edu\/Mathematics\/fmorgan\/Gauss.pdf\">an article<\/a> about this. The formula for how Gauss curvature changes under a conformal change of metric is a simple special case.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>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 : . One can weight arclength and area by [&hellip;]<\/p>\n","protected":false},"author":269,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[14043],"tags":[],"class_list":["post-37","post","type-post","status-publish","format-standard","hentry","category-math"],"acf":[],"_links":{"self":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/37","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/users\/269"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/comments?post=37"}],"version-history":[{"count":1,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/37\/revisions"}],"predecessor-version":[{"id":646,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/37\/revisions\/646"}],"wp:attachment":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/media?parent=37"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/categories?post=37"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/tags?post=37"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}