On isometries of space forms.

*(English)*Zbl 0789.53027
Szenthe, J. (ed.) et al., Differential geometry and its applications. Proceedings of a colloquium, held in Eger, Hungary, August 20-25, 1989, organized by the János Bolyai Mathematical Society. Amsterdam: North- Holland Publishing Company. Colloq. Math. Soc. János Bolyai. 56, 509-534 (1992).

The paper splits into two parts. The first one deals with a well known candidate for a closed hyperbolic 3-manifold of minimal volume (Weeks- Fomenko-Matveev manifold \(M_ 0\), \(\text{vol}M_ 0 = 0.94\dots\)) see A. T. Fomenko and S. V. Matveev [Russ. Math. Surv. 43, No. 1, 3-24 (1988); translation from Usp. Mat. Nauk 43, No. 1, 5-22 (1988; Zbl 0671.58008)]. The author shows that \(M_ 0\) does not regularly cover another closed hyperbolic 3-manifold and its isometry group is a Coxeter group of order 12. The second part studies minimal closed geodesics of closed non-oriented hyperbolic 3-manifolds in some infinite series combinatorically constructed by the author. Arguing to symmetries of corresponding Dirichlet polyhedra, he also describes isometries of these manifolds [see also K. P. Makarova, F. L. Damian and V. V. Balkan, Mat. Issled. 103, 151-163 (1988; Zbl 0669.51012)].

For the entire collection see [Zbl 0764.00002].

For the entire collection see [Zbl 0764.00002].

Reviewer: B.N.Apanasov (Norman)

##### MSC:

53C20 | Global Riemannian geometry, including pinching |

53C22 | Geodesics in global differential geometry |

##### Keywords:

closed hyperbolic 3-manifold; minimal volume; Coxeter group; closed geodesics; Dirichlet polyhedra
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\textit{E. Molnár}, in: Differential geometry and its applications. Proceedings of a colloquium, held in Eger, Hungary, August 20-25, 1989, organized by the János Bolyai Mathematical Society. Amsterdam: North-Holland Publishing Company; Budapest: János Bolyai Mathematical Society. 509--534 (1992; Zbl 0789.53027)