Let $u$ be a radially decreasing function defined on $\mathbb{R}^n$. We consider the metric $g=e^{2u}\delta$ where $\delta$ is the standard Euclidean metric on $\mathbb{R}^n$. Let $B\_r$ be the ball centered at the origin with Euclidean radius $r$. Then for any $x \in \partial B\_r$, by direct computation we know that the line segment connecting the origin and $x$ must be a geodesic. My question is that, is it true that the line segment joining the origin and $x$ is the unique geodesic connecting these two points?


Intuitively this is correct, because such line segment seems to be length minimizing. However, unfortunately I'm not be able to prove this....


I'm actually not very familiar with Riemannian geometry. Any ideas, comments or references will be really appreciated!



Assuming the conformal factor is radially decreasing, prove or disprove the uniqueness of geodesic joining origin and points on the boundary of ball
It must be length minimizing, because it is the unique geodesic from the origin to $x$. In particular, any ray from the origin is geodesic so we know all the geodesics from the origin. Since the shortest path from the origin to $x$ is a geodesic and there is only one such geodesic, it must be length minimizing. 


Note that this does not mean that the shortest path from $x$ to $-x$ is necessarily a line segment through the origin. One could imagine the surface being shaped like a light bulb with the origin being the top of the bulb. If $x$ and $-x$ are on the threading, it's much shorter to go around the threading rather than go to the top and back down. 


