{"id":574,"date":"2008-10-04T13:33:51","date_gmt":"2008-10-04T17:33:51","guid":{"rendered":"http:\/\/people.williams.edu\/morgan\/2008\/10\/04\/paralleltransport\/"},"modified":"2011-06-15T10:15:51","modified_gmt":"2011-06-15T15:15:51","slug":"paralleltransport","status":"publish","type":"post","link":"https:\/\/sites.williams.edu\/Morgan\/2008\/10\/04\/paralleltransport\/","title":{"rendered":"Parallel transport in manifolds with density"},"content":{"rendered":"<p><!--StartFragment--><\/p>\n<p class=\"MsoNormal\">When I spoke on Manifolds with Density (see Chapter 18 of the 2008 edition of my <em><a href=\"http:\/\/www.elsevierdirect.com:80\/product.jsp?isbn=9780123744449\">Geometric Measure Theory<\/a><\/em><span> book)\u00a0<span>\u00a0<\/span>at <a href=\"http:\/\/www.pims.math.ca\/scientific\/past-events\">PIMS<\/a> at the <a href=\"http:\/\/math.ucalgary.ca\/\">University of Calgary<\/a> in September, 2008, <a href=\"http:\/\/math.ucalgary.ca\/profiles\/larry-bates\">Larry Bates<\/a> asked for a generalization of parallel transport to an n-dimensional manifold M with density <img src='https:\/\/s0.wp.com\/latex.php?latex=e%5E%5Cpsi&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='e^\\psi' title='e^\\psi' class='latex' \/>. For n=2, the generalized 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' \/>\u00a0of a curve of Riemannian curvature <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Ckappa_0&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\\kappa_0' title='\\kappa_0' class='latex' \/> involves the log <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Cpsi&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\\psi' title='\\psi' class='latex' \/> of the density as well:<\/span><\/p>\n<p class=\"MsoNormal\" align=\"center\"><span><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' \/><\/span>\u00a0= <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Ckappa_0-%5Cfrac%7Bd%5Cpsi%7D%7Bdn%7D.&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\\kappa_0-\\frac{d\\psi}{dn}.' title='\\kappa_0-\\frac{d\\psi}{dn}.' class='latex' \/>\u00a0<\/p>\n<p class=\"MsoNormal\">Since this describes the rate at which the unit tangent vector is turning, it can be used to define parallel transport.<\/p>\n<p class=\"MsoNormal\"><span>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/span>For n &gt; 2, the generalized curvature vector\u00a0<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' \/>\u00a0should be the Riemannian curvature vector minus the normal component of gradient of the log of the density. Along with the unit tangent, \u00a0<span><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' \/>\u00a0<\/span>determines a plane. Infinitesimally, vectors normal to the plane are transported by classical parallel transport. For vectors in plane, parallel transport is determined by generalized curvature.\u00a0<\/p>\n<p class=\"MsoNormal\"><span>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/span>Are there any applications?<\/p>\n<p><!--EndFragment--><\/p>\n<p class=\"MsoNormal\">\u00a0<\/p>\n<p><!--EndFragment--><\/p>\n","protected":false},"excerpt":{"rendered":"<p>When I spoke on Manifolds with Density (see Chapter 18 of the 2008 edition of my Geometric Measure Theory book)\u00a0\u00a0at PIMS at the University of Calgary in September, 2008, Larry Bates asked for a generalization of parallel transport to an n-dimensional manifold M with density . For n=2, the generalized curvature \u00a0of a curve of [&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-574","post","type-post","status-publish","format-standard","hentry","category-math"],"acf":[],"_links":{"self":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/574","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=574"}],"version-history":[{"count":1,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/574\/revisions"}],"predecessor-version":[{"id":658,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/574\/revisions\/658"}],"wp:attachment":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/media?parent=574"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/categories?post=574"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/tags?post=574"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}