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 Mat. Tr., 2007, Volume 10, Number 2, Pages 3–18 (Mi mt19)

Regular and Quasiregular Isometric Flows on Riemannian Manifolds

V. N. Berestovskiia, Yu. G. Nikonorovb

a Omsk Branch of Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Science
b Rubtsovsk Industrial Intitute, Branch of Altai State Technical University

Abstract: We study the nontrivial Killing vector fields of constant length and the corresponding flows on smooth Riemannian manifolds. We describe the properties of the set of all points of finite (infinite) period for general isometric flows on Riemannian manifolds. It is shown that this flow is generated by an effective almost free isometric action of the group $S^1$ if there are no points of infinite or zero period. In the last case, the set of periods is at most countable and generates naturally an invariant stratification with closed totally geodesic strata; the union of all regular orbits is an open connected dense subset of full measure.

Key words: Riemannian manifold, Killing vector field, action of the circle, geodesic.

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English version:
Siberian Advances in Mathematics, 2008, 18:3, 153–162

Bibliographic databases:

UDC: 514.752.7

Citation: V. N. Berestovskii, Yu. G. Nikonorov, “Regular and Quasiregular Isometric Flows on Riemannian Manifolds”, Mat. Tr., 10:2 (2007), 3–18; Siberian Adv. Math., 18:3 (2008), 153–162

Citation in format AMSBIB
\Bibitem{BerNik07} \by V.~N.~Berestovskii, Yu.~G.~Nikonorov \paper Regular and Quasiregular Isometric Flows on Riemannian Manifolds \jour Mat. Tr. \yr 2007 \vol 10 \issue 2 \pages 3--18 \mathnet{http://mi.mathnet.ru/mt19} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2382415} \transl \jour Siberian Adv. Math. \yr 2008 \vol 18 \issue 3 \pages 153--162 \crossref{https://doi.org/10.3103/S1055134408030012} 

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This publication is cited in the following articles:
1. Berestovskii V.N., Nikitenko E.V., Nikonorov Yu.G., “Classification of generalized normal homogeneous Riemannian manifolds of positive Euler characteristic”, Differential Geom. Appl., 29:4 (2011), 533–546
2. Yu. G. Nikonorov, “Killing vector fields and the curvature tensor of a Riemannian manifold”, Siberian Adv. Math., 24:3 (2014), 187–192
3. Nikonorov Yu.G., “Killing Vector Fields of Constant Length on Compact Homogeneous Riemannian Manifolds”, Ann. Glob. Anal. Geom., 48:4 (2015), 305–330
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