Examples on small cut radius of totally convex set in non-negatively curved manifold - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T22:19:03Z http://mathoverflow.net/feeds/question/119258 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/119258/examples-on-small-cut-radius-of-totally-convex-set-in-non-negatively-curved-manif Examples on small cut radius of totally convex set in non-negatively curved manifold Ralph 2013-01-18T11:58:52Z 2013-01-21T09:45:44Z <p>Suppose $M^n$ is an open complete nonnegatively curved Riemannian manifold. In Cheeger-Gromoll's proof of the soul theorem. They need an estimate on the cut radius of a totally convex set $C$. By a totally convex set $C\in M^n$, we mean for any two point $a, b\in C$ and any geodesic joining them must lie in $A$. In order to define a retraction from $^aC$ to $C$, where $^aC$ is defined by $$^a C=x\in M, d(x, C) &lt; a.$$ Here $a$ must be small enough such that for any $x\in ^aC\setminus C$ there is a unique point $h(a)\in C$ such that $d(x, C)=d(x, h(x))$.</p> <p>I am wondering whether there is a example showing that when $a$ is large, the projection $h$ is not well defined, i.e. there are two points $p, q\in C$ such that $d(x, p)=d(x, q)=d(x, C)$.</p> http://mathoverflow.net/questions/119258/examples-on-small-cut-radius-of-totally-convex-set-in-non-negatively-curved-manif/119276#119276 Answer by Ralf Rueckriemen for Examples on small cut radius of totally convex set in non-negatively curved manifold Ralf Rueckriemen 2013-01-18T15:20:36Z 2013-01-18T15:20:36Z <p>let M be the infinite cylinder ${(x,y,z) | x^2+y^2=1 }$, choose as $C$ a small geodesic disc around some point $p$ and $q$ the point on the opposite site of the circle with the same z coordinate. Then the distance between $q$ and $C$ is realized by two different paths going around the circle in different directions and these paths end in different points in $C$.</p> http://mathoverflow.net/questions/119258/examples-on-small-cut-radius-of-totally-convex-set-in-non-negatively-curved-manif/119452#119452 Answer by Carlo Mantegazza for Examples on small cut radius of totally convex set in non-negatively curved manifold Carlo Mantegazza 2013-01-21T09:45:44Z 2013-01-21T09:45:44Z <p>Let $M$ be the union of the unit northern hemisphere centered at the origin of $R^3$ with the cylinder $\{x^2+y^2=1\,,\, z\leq 0\}$ and $C$ be the geodesic segment $\{x^2+z^2=1\,,\, y=0\,,\, \vert x\vert\leq 1/2\}$. The point $P=(0,1,0)$ is at the same distance from every point of the segment $C$.<br> The metric here is only continuous but it seems to me that "cutting" the hemisphere a little "over" the equator and "gluing" a truncated cone you have a similar example, this time with some point $P$ on the cone which is at equal distance to the two ends of the segment $C$. Moreover, this property should be stable "smoothing" the metric in a thin strip around the gluing circle.</p>