Curvatures of the diagonal lift from an affine manifold to the linear frame bundle

Oldřich Kowalski; Masami Sekizawa

Open Mathematics (2012)

  • Volume: 10, Issue: 3, page 837-843
  • ISSN: 2391-5455

Abstract

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We investigate the curvature of the so-called diagonal lift from an affine manifold to the linear frame bundle LM. This is an affine analogue (but not a direct generalization) of the Sasaki-Mok metric on LM investigated by L.A. Cordero and M. de León in 1986. The Sasaki-Mok metric is constructed over a Riemannian manifold as base manifold. We receive analogous and, surprisingly, even stronger results in our affine setting.

How to cite

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Oldřich Kowalski, and Masami Sekizawa. "Curvatures of the diagonal lift from an affine manifold to the linear frame bundle." Open Mathematics 10.3 (2012): 837-843. <http://eudml.org/doc/269767>.

@article{OldřichKowalski2012,
abstract = {We investigate the curvature of the so-called diagonal lift from an affine manifold to the linear frame bundle LM. This is an affine analogue (but not a direct generalization) of the Sasaki-Mok metric on LM investigated by L.A. Cordero and M. de León in 1986. The Sasaki-Mok metric is constructed over a Riemannian manifold as base manifold. We receive analogous and, surprisingly, even stronger results in our affine setting.},
author = {Oldřich Kowalski, Masami Sekizawa},
journal = {Open Mathematics},
keywords = {Riemannian manifold; Linear frame bundle; Natural metric; Affine connection; Sasaki-Mok metric; linear frame bundle; natural metric; affine connection},
language = {eng},
number = {3},
pages = {837-843},
title = {Curvatures of the diagonal lift from an affine manifold to the linear frame bundle},
url = {http://eudml.org/doc/269767},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Oldřich Kowalski
AU - Masami Sekizawa
TI - Curvatures of the diagonal lift from an affine manifold to the linear frame bundle
JO - Open Mathematics
PY - 2012
VL - 10
IS - 3
SP - 837
EP - 843
AB - We investigate the curvature of the so-called diagonal lift from an affine manifold to the linear frame bundle LM. This is an affine analogue (but not a direct generalization) of the Sasaki-Mok metric on LM investigated by L.A. Cordero and M. de León in 1986. The Sasaki-Mok metric is constructed over a Riemannian manifold as base manifold. We receive analogous and, surprisingly, even stronger results in our affine setting.
LA - eng
KW - Riemannian manifold; Linear frame bundle; Natural metric; Affine connection; Sasaki-Mok metric; linear frame bundle; natural metric; affine connection
UR - http://eudml.org/doc/269767
ER -

References

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  1. [1] Cordero L.A., Dodson C.T.J., de León M., Differential Geometry of Frame Bundles, Math. Appl., 47, Kluwer, Dordrecht, 1989 Zbl0673.53001
  2. [2] Cordero L.A., de León M., On the curvature of the induced Riemannian metric on the frame bundle of a Riemannian manifold, J. Math. Pures Appl., 1986, 65(1), 81–91 Zbl0542.53014
  3. [3] Kolář I., Michor P.W., Slovák J., Natural Operations in Differential Geometry, Springer, Berlin-Heidelberg-New York, 1993 Zbl0782.53013
  4. [4] Kowalski O., Curvature of the induced Riemannian metric on the tangent bundle of a Riemannian manifold, J. Reine Angew. Math., 1971, 250, 124–129 Zbl0222.53044
  5. [5] Kowalski O., Sekizawa M., Natural transformations of Riemannian metrics on manifolds to metrics on linear frame bundles - a classification, In: Differential Geometry and its Applications, Brno, August 24–30, 1986, Math. Appl. (East European Ser.), 27, Reidel, Dordrecht, 1987, 149–178 Zbl0632.53040
  6. [6] Kowalski O., Sekizawa M., On curvatures of linear frame bundles with naturally lifted metrics, Rend. Semin. Mat. Univ. Politec. Torino, 2005, 63(3), 283–295 Zbl1141.53020
  7. [7] Kowalski O., Sekizawa M., Invariance of g-natural metrics on linear frame bundles, Arch. Math. (Brno), 2008, 44(2), 139–147 Zbl1212.53042
  8. [8] Kowalski O., Sekizawa M., On the geometry of orthonormal frame bundles, Math. Nachr., 2008, 281(12), 1799–1809 http://dx.doi.org/10.1002/mana.200610715 Zbl1158.53015
  9. [9] Kowalski O., Sekizawa M., On the geometry of orthonormal frame bundles II, Ann. Global Anal. Geom., 2008, 33(4), 357–371 http://dx.doi.org/10.1007/s10455-007-9091-7 Zbl1141.53023
  10. [10] Kowalski O., Sekizawa M., Invariance of the naturally lifted metrics on linear frame bundles over affine manifolds, Publ. Math. Debrecen (in press), preprint available at http://www.u-gakugei.ac.jp/~sekizawa/Invariance.pdf Zbl1299.53075
  11. [11] Mok K.P., On the differential geometry of frame bundles of Riemannian manifolds, J. Reine Angew. Math., 1978, 302, 16–31 Zbl0378.53016
  12. [12] Musso E., Tricerri F., Riemannian metrics on tangent bundles, Ann. Mat. Pura Appl., 1988, 150, 1–19 http://dx.doi.org/10.1007/BF01761461 Zbl0658.53045
  13. [13] Patterson E.M., Walker A.G., Riemann extensions, Q. J. Math., 1952, 3, 19–28 http://dx.doi.org/10.1093/qmath/3.1.19 
  14. [14] Sekizawa M., Natural transformations of symmetric affine connections on manifolds to metrics on linear frame bundles: a classification, Monatsh. Math., 1988, 105(3), 229–243 http://dx.doi.org/10.1007/BF01636931 Zbl0639.53022
  15. [15] Yano K., Ishihara S., Tangent and Cotangent Bundles: Differential Geometry, Pure Appl. Math., 16, Marcel Dekker, New York, 1973 Zbl0262.53024

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