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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.

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    $\begingroup$ These look like potential homework problems in a 3-manifold theory course. Have you studied any knot theory? $\endgroup$
    – Ryan Budney
    Apr 6, 2020 at 4:08
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    $\begingroup$ @RyanBudney These are some homework problems in our Riemannian geometry course. I know little about knot theory... $\endgroup$
    – Frank Kong
    Apr 6, 2020 at 5:41
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    $\begingroup$ I'm voting to close this question as off-topic because MO is not for helping out with homework or assigned exercises $\endgroup$
    – Yemon Choi
    Apr 6, 2020 at 15:22

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