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2012 | 10 | 3 | 837-843
Tytuł artykułu

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

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EN
Abstrakty
EN
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.
Twórcy
Bibliografia
  • [1] Cordero L.A., Dodson C.T.J., de León M., Differential Geometry of Frame Bundles, Math. Appl., 47, Kluwer, Dordrecht, 1989
  • [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
  • [3] Kolář I., Michor P.W., Slovák J., Natural Operations in Differential Geometry, Springer, Berlin-Heidelberg-New York, 1993
  • [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
  • [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
  • [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
  • [7] Kowalski O., Sekizawa M., Invariance of g-natural metrics on linear frame bundles, Arch. Math. (Brno), 2008, 44(2), 139–147
  • [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
  • [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
  • [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
  • [11] Mok K.P., On the differential geometry of frame bundles of Riemannian manifolds, J. Reine Angew. Math., 1978, 302, 16–31
  • [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
  • [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
  • [15] Yano K., Ishihara S., Tangent and Cotangent Bundles: Differential Geometry, Pure Appl. Math., 16, Marcel Dekker, New York, 1973
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-012-0033-7
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