## Geodesic graphs on special 7-dimensional g.o. manifolds

##
*Zdenek Dusek and Oldrich Kowalski*

**Address.**

Z. Dusek, Department of Algebra and Geometry, Palacky University, Tomkova 40, 779 00 Olomouc, Czech Republic

O. Kowalski, Faculty of Mathematics and Physics, Charles University, Sokolovska 83, 186 75 Prague, Czech Republic

**E-mail. **

dusek@prfnw.upol.cz

kowalski@karlin.mff.cuni.cz

**Abstract.**

In \cite{DK}, the present authors and S. Nik\v cevi\'c constructed the 2-parameter family of invariant Riemannian metrics
on the homogeneous manifolds $M=[{\rm SO}(5)\times{\rm SO}(2)]/{\rm U}(2)$ and $M=[{\rm SO}(4,1)\times{\rm SO}(2)]/{\rm U}(2)$. They proved that, for the open dense subset of this family, the corresponding Riemannian manifolds are g.o. manifolds which are not naturally reductive. Now we are going to investigate the remaining metrics (in the compact case).

**AMSclassification. **22E25, 53C30, 53C35, 53C40.

**Keywords. **Naturally reductive spaces, Riemannian g.o.\ spaces, geodesic graph.