Previous |  Up |  Next

Article

Title: On Riemannian geometry of tangent sphere bundles with arbitrary constant radius (English)
Author: Kowalski, Oldřich
Author: Sekizawa, Masami
Language: English
Journal: Archivum Mathematicum
ISSN: 0044-8753 (print)
ISSN: 1212-5059 (online)
Volume: 44
Issue: 5
Year: 2008
Pages: 391-401
Summary lang: English
.
Category: math
.
Summary: We shall survey our work on Riemannian geometry of tangent sphere bundles with arbitrary constant radius done since the year 2000. (English)
Keyword: Riemannian metric
Keyword: tangent bundle
Keyword: tangent sphere bundle
Keyword: Riemannian curvature
Keyword: scalar curvature
MSC: 53C07
MSC: 53C25
MSC: 53C30
idZBL: Zbl 1212.53043
idMR: MR2501575
.
Date available: 2009-01-29T09:16:03Z
Last updated: 2013-09-19
Stable URL: http://hdl.handle.net/10338.dmlcz/127125
.
Reference: [1] Abbassi, M. T. K., Calvaruso, G.: $g$-natural contact metrics on unit tangent sphere bundles.Monatsh. Math. 151 (2) (2007), 89–109. Zbl 1128.53049, MR 2322938, 10.1007/s00605-006-0421-9
Reference: [2] Adams, J. F.: On the non-existence of elements of Hopf invariant one.Ann. Math. 72 (1960), 20–104. Zbl 0096.17404, MR 0141119, 10.2307/1970147
Reference: [3] Besse, A. L.: Einstein Manifolds.Springer-Verlag, Berlin–Heidelberg–New York, 1987. Zbl 0613.53001, MR 0867684
Reference: [4] Blair, D.: When is the tangent sphere bundle locally symmetric?.Geom. Topol., World Sci. Publishing, Singapore (1989), 15–30. MR 1001586
Reference: [5] Boeckx, E., Vanhecke, L.: Characteristic reflections on unit tangent sphere bundles.Houston J. Math. 23 (1997), 427–448. Zbl 0897.53010, MR 1690045
Reference: [6] Boeckx, E., Vanhecke, L.: Geometry of the tangent sphere bundle.Proceedings of the Workshop on Recent Topics in Differential Geometry (Cordero, L. A., García-Río, E., eds.), Santiago de Compostela, 1997, pp. 5–17.
Reference: [7] Boeckx, E., Vanhecke, L.: Curvature homogeneous unit tangent sphere bundles.Publ. Math. Debrecen 35 (1998), 389–413. MR 1657491
Reference: [8] Boeckx, E., Vanhecke, L.: Unit tangent sphere bundles and two-point homogeneous spaces.Period. Math. Hungar. 36 (1998), 79–95. MR 1694613, 10.1023/A:1004629423529
Reference: [9] Boeckx, E., Vanhecke, L.: Harmonic and minimal vector fields on tangent and unit tangent bundles.Differential Geom. Appl. 13 (2000), 77–93. Zbl 0973.53053, MR 1775222, 10.1016/S0926-2245(00)00021-8
Reference: [10] Boeckx, E., Vanhecke, L.: Unit tangent sphere bundles with constant scalar curvature.Czechoslovak Math. J. 51 (126) (2001), 523–544. Zbl 1079.53063, MR 1851545, 10.1023/A:1013779805244
Reference: [11] Borisenko, A. A., Yampolsky, A. L.: On the Sasaki metric of the tangent and the normal bundles.Sov. Math., Dokl. 35 (1987), 479–482.
Reference: [12] Borisenko, A. A., Yampolsky, A. L.: The sectional curvature of the Sasaki metric of $T_rM^n$.Ukrain. Geom. Sb. 30 (1987), 10–17.
Reference: [13] Borisenko, A. A., Yampolsky, A. L.: Riemannian geometry of fiber bundles.Russian Math. Surveys 46 (6) (1991), 55–106. MR 1164201, 10.1070/RM1991v046n06ABEH002859
Reference: [14] Calvaruso, G.: Contact metric geometry of the unit tangent sphere bundle.Complex, contact and symmetric manifolds. In honor of L. Vanhecke (Kowalski, O. et al, ed.), vol. 234, Progress in Mathematics, 2005, pp. 41–57. Zbl 1079.53045, MR 2105140
Reference: [15] Ivanov, S., Petrova, I.: Riemannian manifold in which the skew-symmetric curvature operator has pointwise constant eigenvalues.Geom. Dedicata 70 (1998), 269–282. Zbl 0903.53016, MR 1624814, 10.1023/A:1005014507809
Reference: [16] Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry II.Interscience Publishers, New York–London–Sydney, 1969. MR 0238225
Reference: [17] Kowalski, O., Sekizawa, M.: Geometry of tangent sphere bundles with arbitrary constant radius.Proceedings of the Symposium Contemporary Mathematics (Bokan, N., ed.), Faculty of Mathematics, University of Belgrade, 2000, pp. 219–228. Zbl 1024.53030, MR 1848571
Reference: [18] Kowalski, O., Sekizawa, M.: On tangent sphere bundles with small or large constant radius.Ann. Global Anal. Geom. 18 (2000), 207–219. Zbl 1011.53025, MR 1795094, 10.1023/A:1006707521207
Reference: [19] Kowalski, O., Sekizawa, M.: On the scalar curvature of tangent sphere bundles with arbitrary constant radius.Bull. Greek Math. Soc. 44 (2000), 17–30. Zbl 1163.53321, MR 1848571
Reference: [20] Kowalski, O., Sekizawa, M.: On Riemannian manifolds whose tangent sphere bundles can have nonnegative sectional curvature.Univ. Jagellon. Acta Math. 40 (2002), 245–256. Zbl 1039.53050, MR 1962729
Reference: [21] Kowalski, O., Sekizawa, M., Vlášek, Z.: Can tangent sphere bundles over Riemannian manifolds have strictly positive sectional curvature?.Global Differential Geometry: The Mathematical Legacy of Alfred Gray (Fernandez, M. and Wolf, J. A., eds.), Contemp. Math. 288 (2001), 110–118. Zbl 1011.53034, MR 1871003, 10.1090/conm/288/04820
Reference: [22] Nagy, P. T.: Geodesics on the tangent sphere bundle of a Riemannian manifold.Geom. Dedicata 7 (1978), 233–243. Zbl 0385.53010, MR 0487892
Reference: [23] Nash, J.: Positive Ricci curvature on fiber bundles.J. Differential Geom. 14 (1979), 241–254. MR 0587552
Reference: [24] Podestà, F.: Isometries of tangent sphere bundles.Boll. Un. Mat. Ital. A(7) 5 (1991), 207–214. MR 1120381
Reference: [25] Poor, W.: Some exotic spheres with positive Ricci curvature.Math. Ann. 216 (1975), 245–252. Zbl 0293.53016, MR 0400110, 10.1007/BF01430964
Reference: [26] Takagi, H.: Conformally flat Riemannian manifolds admitting a transitive group of isometries.Tôhoku Math. J. 27 (1975), 103–110. Zbl 0323.53037, MR 0442852, 10.2748/tmj/1178241040
Reference: [27] Wolf, J. A.: Elliptic spaces in Grassmann manifolds.Illinois J. Math. 7 (1963), 447–462. MR 0156295
Reference: [28] Yampolsky, A. L.: On the geometry of tangent sphere bundles of Riemannian manifolds.Ukrain. Geom. Sb 24 (1981), 129–132, in Russian. MR 0629822
Reference: [29] Yampolsky, A. L.: On Sasaki metric of tangent and normal bundle.Ph.D. thesis, Odessa, 1986, (Russian).
.

Files

Files Size Format View
ArchMathRetro_044-2008-5_6.pdf 486.7Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo