# Example of compact manifold with bdd such that there exists geodesic whose first time of hitting boundary is different form exit time of geodesic,

Let M be compact Riemann manifold with boundary.

The unit sphere bundle $$SM$$ is given by $$SM=\{(x, v)||v|_{g}=1,x \in M\}$$ where $$g$$ is the Riemannian metric in the tangent space at $$x$$.

Given $$(x, v) \in S M$$, let $$\gamma_{x, v}$$ denote the unique geodesic determined by $$(x, v)$$ so that $$\gamma_{x, v}(0)=x$$ and $$\dot{\gamma}_{x, v}(0)=v$$. For any $$(x, v) \in S M$$ the geodesic $$\gamma_{x, v}$$ is defined on a maximal interval of existence that we denote by $$\left[-\tau_{-}(x, v), \tau_{+}(x, v)\right]$$ where $$\tau_{\pm}(x, v) \in[0, \infty]$$, so that $$\gamma_{x, v}:\left[-\tau_{-}(x, v), \tau_{+}(x, v)\right] \rightarrow M$$ is a smooth curve that cannot be extended to any larger interval as a smooth curve in $$M$$.

We let $$\tau(x, v):=\tau_{+}(x, v)$$ Thus $$\tau(x, v)$$ is the exit time when the geodesic $$\gamma_{x, v}$$ exits $$M$$.

I am interested in finding an example of manifold such that there exists geodesic at point $$x$$ and direction $$v$$ such that first time geodesic to hit the boundary is different from exit time of geodesic $$\tau_{(x,v)}$$.

I could not imagine an example where geodesic reach boundary but come back again inside the manifold.

Any help or hint will be appreciated.

Consider a sphere with a small open disk removed.

Then it is a manifold with boundary, the boundary being a circle. Consider a geodesic starting from a point on this circle and going tangentially to the circle boundary.

Then this geodesic is defined on $$\mathbb{R}$$ but hits the boundary periodically.

The degenerate case is when the boudary circle is a great circle: in this case, the boundary is totally geodesic, and the geodesic constructed above stays forever in the boundary while being defined for all time.

Another example, this time with a finite exit time:

This is a flat torus with a little open disk removed.

• I have one doubt. As you said geodesic starting at a point on the circle which is already a boundary, traveling along tangentially. I know that geodesics of spheres are only great circles. So what will be the trajectory of that geodesics? Can you please through some light? Is just moving along circle boundary? Commented Apr 29, 2021 at 6:29
• What I said is that if the boundary is little enough, there is a tangent great circle to the little enough boundary which does not goes out of the manifold. Commented Apr 29, 2021 at 6:32
• @idon'tknow I added another example with a finite exit time that is not equal to the first boundary hit Commented Apr 29, 2021 at 7:02
• Thanks a lot for adding the figure. Now it is clear to me. Commented Apr 29, 2021 at 16:09