A class of metrics on tangent bundles of pseudo-Riemannian manifolds

H. M. Dida; A. Ikemakhen

Archivum Mathematicum (2011)

  • Volume: 047, Issue: 4, page 293-308
  • ISSN: 0044-8753

Abstract

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We provide the tangent bundle T M of pseudo-Riemannian manifold ( M , g ) with the Sasaki metric g s and the neutral metric g n . First we show that the holonomy group H s of ( T M , g s ) contains the one of ( M , g ) . What allows us to show that if ( T M , g s ) is indecomposable reducible, then the basis manifold ( M , g ) is also indecomposable-reducible. We determine completely the holonomy group of ( T M , g n ) according to the one of ( M , g ) . Secondly we found conditions on the base manifold under which ( T M , g s ) ( respectively ( T M , g n ) ) is Kählerian, locally symmetric or Einstein manifolds. ( T M , g n ) is always reducible. We show that it is indecomposable if ( M , g ) is irreducible.

How to cite

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Dida, H. M., and Ikemakhen, A.. "A class of metrics on tangent bundles of pseudo-Riemannian manifolds." Archivum Mathematicum 047.4 (2011): 293-308. <http://eudml.org/doc/247005>.

@article{Dida2011,
abstract = {We provide the tangent bundle $TM$ of pseudo-Riemannian manifold $(M,g)$ with the Sasaki metric $g^s$ and the neutral metric $g^n$. First we show that the holonomy group $H^s$ of $(TM ,g^s)$ contains the one of $(M,g)$. What allows us to show that if $(TM ,g^s)$ is indecomposable reducible, then the basis manifold $(M,g)$ is also indecomposable-reducible. We determine completely the holonomy group of $(TM ,g^n)$ according to the one of $(M,g)$. Secondly we found conditions on the base manifold under which $(TM ,g^s)$ ( respectively $(TM ,g^n)$ ) is Kählerian, locally symmetric or Einstein manifolds. $(TM ,g^n)$ is always reducible. We show that it is indecomposable if $(M,g)$ is irreducible.},
author = {Dida, H. M., Ikemakhen, A.},
journal = {Archivum Mathematicum},
keywords = {pseudo-Riemannian manifold; tangent bundle; Sasaki metric; neutral metric; holonomy group; indecomposable-reducible manifold; Einstein manifold; pseudo-Riemannian manifold; Sasaki metric; neutral metric; indecomposable-reducible manifold; Einstein manifold},
language = {eng},
number = {4},
pages = {293-308},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {A class of metrics on tangent bundles of pseudo-Riemannian manifolds},
url = {http://eudml.org/doc/247005},
volume = {047},
year = {2011},
}

TY - JOUR
AU - Dida, H. M.
AU - Ikemakhen, A.
TI - A class of metrics on tangent bundles of pseudo-Riemannian manifolds
JO - Archivum Mathematicum
PY - 2011
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 047
IS - 4
SP - 293
EP - 308
AB - We provide the tangent bundle $TM$ of pseudo-Riemannian manifold $(M,g)$ with the Sasaki metric $g^s$ and the neutral metric $g^n$. First we show that the holonomy group $H^s$ of $(TM ,g^s)$ contains the one of $(M,g)$. What allows us to show that if $(TM ,g^s)$ is indecomposable reducible, then the basis manifold $(M,g)$ is also indecomposable-reducible. We determine completely the holonomy group of $(TM ,g^n)$ according to the one of $(M,g)$. Secondly we found conditions on the base manifold under which $(TM ,g^s)$ ( respectively $(TM ,g^n)$ ) is Kählerian, locally symmetric or Einstein manifolds. $(TM ,g^n)$ is always reducible. We show that it is indecomposable if $(M,g)$ is irreducible.
LA - eng
KW - pseudo-Riemannian manifold; tangent bundle; Sasaki metric; neutral metric; holonomy group; indecomposable-reducible manifold; Einstein manifold; pseudo-Riemannian manifold; Sasaki metric; neutral metric; indecomposable-reducible manifold; Einstein manifold
UR - http://eudml.org/doc/247005
ER -

References

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