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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].
Reviewer: O.Kowalski (Praha)

##### MSC:
 53C30 Differential geometry of homogeneous manifolds 53C22 Geodesics in global differential geometry