Let $G= \left\{ \begin{pmatrix} 1&a&c\\0&1&b\\0&0&0 \end{pmatrix} \mid a,b,c\in \mathbb{R} \right\}$ be the Heisenberg group. Is there a compact codimension one submanifold of $G$ which is totally geodesic? (We fix a left invariant metric for the Heisenberg group)



totally geodesic submanifold of Heisenberg group
The question is studied in detail in [Pat Eberlein's 1994 paper.](http://www.ams.org/journals/tran/1994-343-02/S0002-9947-1994-1250818-2/S0002-9947-1994-1250818-2.pdf) I believe the answer to your specific question is **NO.**


