Riemannian manifolds of cohomogeneity one.

*(English)*Zbl 0795.53044
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, 9-22 (1992).

The present author defined in 1979 a cohomogeneity one Riemannian manifold, shortly a \(C1\) manifold, as a \(G\)-manifold with at least one orbit of codimension one and admitting a complete \(G\)-invariant Riemannian metric. Later, L. Berárd-Bergery showed that the orbit space of a \(C1\) manifold is either \(R\), or \(S^ 1\), or \(R^ +\), or a closed interval. In the first two cases, all orbits have codimension one; in the third case there is one singular orbit and in the last case there are two singular orbits. In the present work, the author describes smooth invariant metrics on \(C1\) manifolds and studies geodesics which are normal to the orbits. The cases with none, one or two singular orbits are investigated separately. A large number of interesting results is given with comments but without proofs.

For the entire collection see [Zbl 0764.00002].

For the entire collection see [Zbl 0764.00002].

Reviewer: O.Kowalski (Praha)

##### MSC:

53C30 | Differential geometry of homogeneous manifolds |

53C22 | Geodesics in global differential geometry |

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\textit{D. V. Alekseevsky}, 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. 9--22 (1992; Zbl 0795.53044)