Injectivity radius and the cut locus - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T02:24:20Z http://mathoverflow.net/feeds/question/36082 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/36082/injectivity-radius-and-the-cut-locus Injectivity radius and the cut locus Dror Atariah 2010-08-19T13:41:55Z 2010-08-20T14:09:56Z <p>Consider a connected, complete and compact Riemannian manifold $M$. Is it correct that the following equality holds: $\text{inj}(x)=\text{dist}\left(x,\text{CuL}(x)\right)$? Or in words that the injectivity radius of a point is the distance from the point to its cut locus.</p> <p>Here is my explanation: As the manifold is compact and complete, then the cut locus $\text{CuL}(x)$ is compact as well[1]. Thus, there exists a point $y\in \text{CuL}(x)$ such that $\text{dist}\left(x,\text{CuL}(x)\right)=\text{dist}(x,y)$. Since $y$ is a cut point of $x$, there exists a tangent vector $\xi_0\in T_x M$ such that $y=\exp_x\left(c(\xi_0)\xi_0\right)$[2], where $c(\xi_0)$ is the distance from $x$ to its cut point in the $\xi_0$ direction. This in turn means that $\text{dist}(x,y) = c(\xi_0)$. </p> <p>Recall that $\text{inj}(x)=\inf_{\xi\in T_x M}(c(\xi))$. This means that $\text{inj}(x) \leq c(\xi_0) = \text{dist}(x,y)=\text{dist}\left(x,\text{CuL}(x)\right)$. If $\text{inj}(x)&lt; c(\xi_0)$, then since $M$ is compact, it means that there exists some other tangent vector $\xi\in T_x M$ with $c(\xi) &lt; c(\xi_0)$. But this means that $\exp_x(c(\xi)\xi)$ is a cut point of $x$ closer to it then $y$, and this is a contradiction.</p> <hr> <p>[1] See <em>Contributions to Riemannian Geometry in the Large</em> by W. Klingenberg</p> <p>[2] Here I'm using the notation of I. Chavel in his book <em>Riemannian Geometry - Modern Introduction</em>.</p> <hr> <p><strong>Update(@dror)</strong> Today I finally found a copy of the book *Riemannian Geometry" by Takashi Sakai, and there the above is stated as proposition 4.13 in chapter 3. Thanks anyway.</p> http://mathoverflow.net/questions/36082/injectivity-radius-and-the-cut-locus/36083#36083 Answer by Deane Yang for Injectivity radius and the cut locus Deane Yang 2010-08-19T14:51:57Z 2010-08-19T14:51:57Z <p>The injectivity radius for a point $x$ is the largest distance $r$ such that any geodesic starting from x is length-minimizing for at least distance $r$. So there exists at least one geodesic starting from $x$ that is <em>not</em> length-minimizing past distance $r$. On the other hand a point $p$ is in the cut locus of $x$ if a geodesic starting from $x$ and passing through $p$ is <em>not</em> length-minimizing for any point past $p$. So the injectivity radius at $x$ is the distance from $x$ to its cut locus.</p>