Injectivity radius and the cut locus

2010-08-19T13:41:55Z

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.

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 $\text{dist}(x,y) = c(\xi_0)$.

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 $\text{inj}(x)< c(\xi_0)$, then since $M$ is compact, it means that there exists some other tangent vector $\xi\in T_x M$ with $c(\xi) < c(\xi_0)$. 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.

[1] See Contributions to Riemannian Geometry in the Large by W. Klingenberg

[2] Here I'm using the notation of I. Chavel in his book Riemannian Geometry - Modern Introduction.

Update(@dror) 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.

Answer by Deane Yang for Injectivity radius and the cut locus

2010-08-19T14:51:57Z

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 not 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 not length-minimizing for any point past $p$. So the injectivity radius at $x$ is the distance from $x$ to its cut locus.

Injectivity radius and the cut locus - MathOverflow

Injectivity radius and the cut locus

Answer by Deane Yang for Injectivity radius and the cut locus