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Sibirsk. Mat. Zh., 2008, Volume 49, Number 3, Pages 497–514 (Mi smj1856)  

This article is cited in 18 scientific papers (total in 18 papers)

Killing vector fields of constant length 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 complete smooth Riemannian manifolds. Various examples are constructed of the Killing vector fields of constant length generated by the isometric effective almost free but not free actions of $S^1$ on the Riemannian manifolds close in some sense to symmetric spaces. The latter manifolds include “almost round” odd-dimensional spheres and unit vector bundles over Riemannian manifolds. We obtain some curvature constraints on the Riemannian manifolds admitting nontrivial Killing fields of constant length.

Keywords: Riemannian manifold, Killing vector field, geodesic, Sasaki metric.

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English version:
Siberian Mathematical Journal, 2008, 49:3, 395–407

Bibliographic databases:

UDC: 514.752.7
Received: 08.12.2006

Citation: V. N. Berestovskii, Yu. G. Nikonorov, “Killing vector fields of constant length on Riemannian manifolds”, Sibirsk. Mat. Zh., 49:3 (2008), 497–514; Siberian Math. J., 49:3 (2008), 395–407

Citation in format AMSBIB
\by V.~N.~Berestovskii, Yu.~G.~Nikonorov
\paper Killing vector fields of constant length on Riemannian manifolds
\jour Sibirsk. Mat. Zh.
\yr 2008
\vol 49
\issue 3
\pages 497--514
\jour Siberian Math. J.
\yr 2008
\vol 49
\issue 3
\pages 395--407

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    This publication is cited in the following articles:
    1. Nascimento A.S., Goncalves A.C., “Instability of Elliptic Equations on Compact Riemannian Manifolds with Non-Negative Ricci Curvature”, Electronic Journal of Differential Equations, 2010, 67  mathscinet  zmath  isi
    2. 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  crossref  mathscinet  isi  elib  scopus
    3. Deshmukh Sh., “Characterizations of Einstein manifolds and odd-dimensional spheres”, J Geom Phys, 61:11 (2011), 2058–2063  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    4. Deshmukh Sh., “Jacobi-type vector fields on Ricci solitons”, Bulletin Mathematique de La Societe Des Sciences Mathematiques de Roumanie, 55:1 (2012), 41–50  mathscinet  zmath  isi
    5. Deshmukh Sh., “A Note on Compact Hypersurfaces in a Euclidean Space”, C. R. Math., 350:21-22 (2012), 971–974  crossref  mathscinet  zmath  isi  elib  scopus
    6. Souam R., Van der Veken J., “Totally Umbilical Hypersurfaces of Manifolds Admitting a Unit Killing Field”, Trans. Am. Math. Soc., 364:7 (2012), 3609–3626  crossref  mathscinet  zmath  isi  scopus
    7. Yu. G. Nikonorov, “Killing vector fields and the curvature tensor of a Riemannian manifold”, Siberian Adv. Math., 24:3 (2014), 187–192  mathnet  crossref  mathscinet
    8. Gimeno V., Sotoca J.M., “Geometric Approach to Non-Relativistic Quantum Dynamics of Mixed States”, J. Math. Phys., 54:5 (2013), 052108  crossref  mathscinet  zmath  isi  elib  scopus
    9. Nikonorov Yu.G., “Geodesic Orbit Manifolds and Killing Fields of Constant Length”, Hiroshima Math. J., 43:1 (2013), 129–137  mathscinet  zmath  isi  elib
    10. Berestovskii V.N. Nikonorov Yu.G., “Generalized Normal Homogeneous Riemannian Metrics on Spheres and Projective Spaces”, Ann. Glob. Anal. Geom., 45:3 (2014), 167–196  crossref  mathscinet  zmath  isi  elib  scopus
    11. Olea B., “Canonical Variation of a Lorentzian Metric”, J. Math. Anal. Appl., 419:1 (2014), 156–171  crossref  mathscinet  zmath  isi  elib  scopus
    12. Yoshikawa R., Sabau S.V., “Kropina Metrics and Zermelo Navigation on Riemannian Manifolds”, Geod. Dedic., 171:1 (2014), 119–148  crossref  mathscinet  zmath  isi  scopus
    13. Nikonorov Yu.G., “Killing Vector Fields of Constant Length on Compact Homogeneous Riemannian Manifolds”, Ann. Glob. Anal. Geom., 48:4 (2015), 305–330  crossref  mathscinet  zmath  isi  elib  scopus
    14. Shenawy S., Unal B., “2-Killing Vector Fields on Warped Product Manifolds”, Int. J. Math., 26:9 (2015), 1550065  crossref  mathscinet  zmath  isi  elib  scopus
    15. Aazami A.B., “the Newman-Penrose Formalism For Riemannian 3-Manifolds”, J. Geom. Phys., 94 (2015), 1–7  crossref  mathscinet  zmath  isi  elib  scopus
    16. Di Scala A.J., “Killing Vector Fields of Constant Length on Compact Hypersurfaces”, Geod. Dedic., 175:1 (2015), 403–406  crossref  mathscinet  zmath  isi  scopus
    17. Dirmeier A., Plaue M., Scherfner M., “on the Causal Structure of Stationary Spacetimes in Standard Form With Compact Fiber”, J. Geom. Phys., 110 (2016), 277–281  crossref  mathscinet  zmath  isi  scopus
    18. El-Sayied H.K., Shenawy S., Syied N., “Conformal Vector Fields on Doubly Warped Product Manifolds and Applications”, Adv. Math. Phys., 2016, 6508309  crossref  mathscinet  zmath  isi  scopus
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