# totally geodesic submanifold of Heisenberg group

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)

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The question is studied in detail in Pat Eberlein's 1994 paper. I believe the answer to your specific question is NO.

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thank you very much for your very interesting link. –  Ali Taghavi Jul 6 '14 at 20:02