Invariance of g -natural metrics on linear frame bundles

Oldřich Kowalski; Masami Sekizawa

Archivum Mathematicum (2008)

  • Volume: 044, Issue: 2, page 139-147
  • ISSN: 0044-8753

Abstract

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In this paper we prove that each g -natural metric on a linear frame bundle L M over a Riemannian manifold ( M , g ) is invariant with respect to a lifted map of a (local) isometry of the base manifold. Then we define g -natural metrics on the orthonormal frame bundle O M and we prove the same invariance result as above for O M . Hence we see that, over a space ( M , g ) of constant sectional curvature, the bundle O M with an arbitrary g -natural metric G ˜ is locally homogeneous.

How to cite

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Kowalski, Oldřich, and Sekizawa, Masami. "Invariance of $g$-natural metrics on linear frame bundles." Archivum Mathematicum 044.2 (2008): 139-147. <http://eudml.org/doc/250437>.

@article{Kowalski2008,
abstract = {In this paper we prove that each $g$-natural metric on a linear frame bundle $LM$ over a Riemannian manifold $(M, g)$ is invariant with respect to a lifted map of a (local) isometry of the base manifold. Then we define $g$-natural metrics on the orthonormal frame bundle $OM$ and we prove the same invariance result as above for $OM$. Hence we see that, over a space $(M, g)$ of constant sectional curvature, the bundle $OM$ with an arbitrary $g$-natural metric $\tilde\{G\}$ is locally homogeneous.},
author = {Kowalski, Oldřich, Sekizawa, Masami},
journal = {Archivum Mathematicum},
keywords = {Riemannian manifold; linear frame bundle; orthonormal frame bundle; $g$-natural metrics; homogeneity; Riemannian manifold; linear frame bundle; orthonormal frame bundle; -natural metric; homogeneity},
language = {eng},
number = {2},
pages = {139-147},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Invariance of $g$-natural metrics on linear frame bundles},
url = {http://eudml.org/doc/250437},
volume = {044},
year = {2008},
}

TY - JOUR
AU - Kowalski, Oldřich
AU - Sekizawa, Masami
TI - Invariance of $g$-natural metrics on linear frame bundles
JO - Archivum Mathematicum
PY - 2008
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 044
IS - 2
SP - 139
EP - 147
AB - In this paper we prove that each $g$-natural metric on a linear frame bundle $LM$ over a Riemannian manifold $(M, g)$ is invariant with respect to a lifted map of a (local) isometry of the base manifold. Then we define $g$-natural metrics on the orthonormal frame bundle $OM$ and we prove the same invariance result as above for $OM$. Hence we see that, over a space $(M, g)$ of constant sectional curvature, the bundle $OM$ with an arbitrary $g$-natural metric $\tilde{G}$ is locally homogeneous.
LA - eng
KW - Riemannian manifold; linear frame bundle; orthonormal frame bundle; $g$-natural metrics; homogeneity; Riemannian manifold; linear frame bundle; orthonormal frame bundle; -natural metric; homogeneity
UR - http://eudml.org/doc/250437
ER -

References

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