# 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

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topOldřich 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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- [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
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- [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] 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] Mok K.P., On the differential geometry of frame bundles of Riemannian manifolds, J. Reine Angew. Math., 1978, 302, 16–31 Zbl0378.53016
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- [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] 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] 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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