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V. N. Berestovskii Yu. G. Nikonorov Rubtsovsk Industrial Institute of Altai State Technical University Rubtsovsk, Russia nik@inst.rubtsovsk.ru

On Killing vector fields of constant length on homogeneous Riemannian manifolds
The main ob ject of exploration in this talk are Kil ling vector fields of constant length on homogeneous Riemannian manifolds. A Killing vector field X on a Riemannian manifold (M , g ) has constant length if and only if every integral curve of the field X is a geodesic in (M , g ). It is known a connections between such fields and one-parameter groups (t), t R, (flows) of Clifford-Wolf translations (i.e. isometries moving all points one and the same distance) on a given Riemannian manifold (M , g ). First of all, every such flow is generated by a Killing vector field X of constant length on (M , g ). If X is a unit Killing vector field on a compact (or on a homogeneous) Riemannian manifold M , then all isometry from the one-parameter group (t) generated by X (t R ) with sufficiently small |t| are Clifford-Wolf translations on M . This property is not valid in general for any t R . It is remarkable that the flow on symmetric space, generated by a unit Killing vector field, consists of Clifford-Wolf translations [1]. It is a hard problem to obtain a detailed description of Killing vector fields of constant length on arbitrary homogeneous Riemannian manifold. In the second part of the talk we discuss some recent results in this direction. In particular, the following Theorem. Let G be a transitive semisimple compact isometry group of a Riemannian manifold M . Suppose that there exists a Kil ling field X g of constant length, which is a regular element of g, where g is the Lie algebra of G. Then al l Y g are of constant length, and M is isometric to G supplied with a bi-invariant Riemannian metric. Acknowlegments. The pro ject was supported in part by RFBR (grant N 05-01-00611a) and the State Maintenance Program for the Leading Scientific Schools and Young Russian Scientists of the Russian Federation (grants NSH-8526.2006.1 and MD-5179.2006.1). References [1] V.N. Berestovskii and Yu.G. Nikonorov, Killing vector fields of constant length on Riemannian manifolds, Transformation Groups, to appear.

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