{"version":"1.0","provider_name":"SOUL OF MATHEMATICS","provider_url":"https:\/\/soulofmathematics.com","author_name":"Rajarshi Dey","author_url":"https:\/\/soulofmathematics.com\/index.php\/author\/rajarshidey1729gmail-com\/","title":"THE SOUL THEOREM - SOUL OF MATHEMATICS","type":"rich","width":600,"height":338,"html":"<blockquote class=\"wp-embedded-content\" data-secret=\"XgtxgKGjAv\"><a href=\"https:\/\/soulofmathematics.com\/index.php\/the-soul-theorem\/\">THE SOUL THEOREM<\/a><\/blockquote><iframe sandbox=\"allow-scripts\" security=\"restricted\" src=\"https:\/\/soulofmathematics.com\/index.php\/the-soul-theorem\/embed\/#?secret=XgtxgKGjAv\" width=\"600\" height=\"338\" title=\"&#8220;THE SOUL THEOREM&#8221; &#8212; SOUL OF MATHEMATICS\" data-secret=\"XgtxgKGjAv\" frameborder=\"0\" marginwidth=\"0\" marginheight=\"0\" scrolling=\"no\" class=\"wp-embedded-content\"><\/iframe><script type=\"text\/javascript\">\n\/* <![CDATA[ *\/\n\/*! This file is auto-generated *\/\n!function(d,l){\"use strict\";l.querySelector&&d.addEventListener&&\"undefined\"!=typeof URL&&(d.wp=d.wp||{},d.wp.receiveEmbedMessage||(d.wp.receiveEmbedMessage=function(e){var t=e.data;if((t||t.secret||t.message||t.value)&&!\/[^a-zA-Z0-9]\/.test(t.secret)){for(var s,r,n,a=l.querySelectorAll('iframe[data-secret=\"'+t.secret+'\"]'),o=l.querySelectorAll('blockquote[data-secret=\"'+t.secret+'\"]'),c=new RegExp(\"^https?:$\",\"i\"),i=0;i<o.length;i++)o[i].style.display=\"none\";for(i=0;i<a.length;i++)s=a[i],e.source===s.contentWindow&&(s.removeAttribute(\"style\"),\"height\"===t.message?(1e3<(r=parseInt(t.value,10))?r=1e3:~~r<200&&(r=200),s.height=r):\"link\"===t.message&&(r=new URL(s.getAttribute(\"src\")),n=new URL(t.value),c.test(n.protocol))&&n.host===r.host&&l.activeElement===s&&(d.top.location.href=t.value))}},d.addEventListener(\"message\",d.wp.receiveEmbedMessage,!1),l.addEventListener(\"DOMContentLoaded\",function(){for(var e,t,s=l.querySelectorAll(\"iframe.wp-embedded-content\"),r=0;r<s.length;r++)(t=(e=s[r]).getAttribute(\"data-secret\"))||(t=Math.random().toString(36).substring(2,12),e.src+=\"#?secret=\"+t,e.setAttribute(\"data-secret\",t)),e.contentWindow.postMessage({message:\"ready\",secret:t},\"*\")},!1)))}(window,document);\n\/* ]]> *\/\n<\/script>\n","thumbnail_url":"https:\/\/i1.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2020\/08\/unnamed.gif?fit=500%2C500&ssl=1","thumbnail_width":500,"thumbnail_height":500,"description":"We think mathematics to be a subject too coarse to have a connection to any spirit, let alone to have its own. But Mathematicians are probably the only people to have named a theorem &#8216;THE SOUL THEOREM&#8217;. The&nbsp;soul theorem&nbsp;is a theorem of&nbsp;Riemannian geometry&nbsp;that largely reduces the study of complete&nbsp;manifolds&nbsp;of non-negative&nbsp;sectional curvature&nbsp;to that of the&nbsp;compact&nbsp;case. Every&nbsp;compact&nbsp;manifold is its own soul. In 1972, Cheeger&nbsp;and&nbsp;Gromoll&nbsp;proved the theorem by the generalization of a 1969 result of Gromoll and Wolfgang Meyer. The related&nbsp;soul conjecture&nbsp;was formulated by Gromoll and Cheeger in 1972 and proved by&nbsp;Grigori Perelman&nbsp;in 1994 with an astonishingly concise proof. The theorem states, If&nbsp;(M,&nbsp;g)&nbsp;is a&nbsp;complete connected Riemannian manifold&nbsp;with&nbsp;sectional curvature K&nbsp;\u2265 0, then there exists a&nbsp;compact totally convex,&nbsp;totally geodesic submanifold S&nbsp;whose&nbsp;normal bundle&nbsp;is&nbsp;diffeomorphic&nbsp;to&nbsp;M. (Note that the sectional curvature must be non-negative everywhere, but it does not have to be constant.) Such a submanifold&nbsp;S&nbsp;is called a&nbsp;soul&nbsp;of&nbsp;(M,&nbsp;g). SOUL CONJECTURE The Cheeger and Gromoll&#8217;s&nbsp;soul conjecture&nbsp;states, Suppose&nbsp;(M,&nbsp;g)&nbsp;is complete, connected and non-compact with sectional curvature&nbsp;K&nbsp;\u2265 0, and there exists a point in&nbsp;M&nbsp;where the sectional curvature (in all sectional directions) is strictly positive. Then the soul of&nbsp;M&nbsp;is a point; equivalently&nbsp;M&nbsp;is diffeomorphic to&nbsp;Rn. Grigori Perelman&nbsp; established that in the general case&nbsp;K&nbsp;\u2265 0,&nbsp;Sharafutdinov&#8217;s retraction&nbsp;P&nbsp;: M \u2192 S&nbsp;is a&nbsp;submersion and hence proved the soul conjecture. In this note we consider complete noncompact Riemannian manifolds M of nonnegative sectional curvature. The structure of such manifolds was discovered by Cheeger and Gromoll : M contains a (not necessarily unique) totally convex and totally geodesic submanifold S without boundary, 0 &lt; dimS &lt; dimM, such that M is diffeomorphic to the total space of the normal bundle of S in M . (S is called a soul of M.) In particular, if S is a single point, then M is diffeomorphic to a Euclidean space. This is the case if all sectional curvatures of M are positive, according to an earlier result of Gromoll and Meyer. Cheeger and Gromoll conjectured that the same conclusion can be obtained under the weaker assumption that M contains a point where all sectional curvatures are positive. A contrapositive version of this conjecture expresses certain rigidity of manifolds with souls of positive dimension. It was verified in the cases dim S = 1 and codimS = 1, and by Marenich, Walschap, and Strake in the case codimS = 2. EXAMPLE, As a very simple example, take&nbsp;M&nbsp;to be&nbsp;Euclidean space&nbsp;Rn. The sectional curvature is&nbsp;0&nbsp;everywhere, and any point of&nbsp;M&nbsp;can serve as a soul of&nbsp;M. Now take the&nbsp;paraboloid&nbsp;M&nbsp;= {(x,&nbsp;y,&nbsp;z)&nbsp;:&nbsp;z&nbsp;=&nbsp;x2&nbsp;+&nbsp;y2}, with the metric&nbsp;g&nbsp;being the ordinary Euclidean distance coming from the embedding of the paraboloid in Euclidean space&nbsp;R3. Here the sectional curvature is positive everywhere, though not constant. The origin&nbsp;(0, 0, 0)&nbsp;is a soul of&nbsp;M. Not every point&nbsp;x&nbsp;of&nbsp;M&nbsp;is a soul of&nbsp;M, since there may be geodesic loops based at&nbsp;x, in which case &nbsp;wouldn&#8217;t be totally convex. Citation Perelman, G. Proof of the soul conjecture of Cheeger and Gromoll. J. Differential Geom. 40 (1994), no. 1, 209&#8211;212. doi:10.4310\/jdg\/1214455292."}