# Find a manifold with boundary of a geodesic ball being a torus [closed]

I would like to find the answers to the following questions:

a. Find a complete $$3$$-dimensional Riemannian manifold $$M$$ and a point $$p\in M$$, such that the boundary of the open geodesic ball $$B(p,1)$$ is a smooth $$2$$-dimensional manifold and is diffeomorphic to the torus $$T^2$$.

b. Is there a complete $$3$$-dimensional Riemannian manifold $$M$$ and a point $$p\in M$$, such that the boundary of the open geodesic ball $$B(p,r)$$ for some $$r>0$$ is a smooth $$2$$-dimensional manifold and is isometric to the flat torus $$T^2$$? (The flat torus is defined as the direct product of two copies of $$\mathbb{R}/\mathbb{Z}$$)

Thanks for any help to the questions in advance.

• These look like potential homework problems in a 3-manifold theory course. Have you studied any knot theory? – Ryan Budney Apr 6 at 4:08
• @RyanBudney These are some homework problems in our Riemannian geometry course. I know little about knot theory... – Frank Kong Apr 6 at 5:41
• I'm voting to close this question as off-topic because MO is not for helping out with homework or assigned exercises – Yemon Choi Apr 6 at 15:22