On Riemannian geometry of tangent sphere bundles with arbitrary constant radius

Oldřich Kowalski; Masami Sekizawa

Archivum Mathematicum (2008)

  • Volume: 044, Issue: 5, page 391-401
  • ISSN: 0044-8753

Abstract

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We shall survey our work on Riemannian geometry of tangent sphere bundles with arbitrary constant radius done since the year 2000.

How to cite

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Kowalski, Oldřich, and Sekizawa, Masami. "On Riemannian geometry of tangent sphere bundles with arbitrary constant radius." Archivum Mathematicum 044.5 (2008): 391-401. <http://eudml.org/doc/250504>.

@article{Kowalski2008,
abstract = {We shall survey our work on Riemannian geometry of tangent sphere bundles with arbitrary constant radius done since the year 2000.},
author = {Kowalski, Oldřich, Sekizawa, Masami},
journal = {Archivum Mathematicum},
keywords = {Riemannian metric; tangent bundle; tangent sphere bundle; Riemannian curvature; scalar curvature; Riemannian metric; tangent bundle; tangent sphere bundle; Riemannian curvature; scalar curvature},
language = {eng},
number = {5},
pages = {391-401},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On Riemannian geometry of tangent sphere bundles with arbitrary constant radius},
url = {http://eudml.org/doc/250504},
volume = {044},
year = {2008},
}

TY - JOUR
AU - Kowalski, Oldřich
AU - Sekizawa, Masami
TI - On Riemannian geometry of tangent sphere bundles with arbitrary constant radius
JO - Archivum Mathematicum
PY - 2008
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 044
IS - 5
SP - 391
EP - 401
AB - We shall survey our work on Riemannian geometry of tangent sphere bundles with arbitrary constant radius done since the year 2000.
LA - eng
KW - Riemannian metric; tangent bundle; tangent sphere bundle; Riemannian curvature; scalar curvature; Riemannian metric; tangent bundle; tangent sphere bundle; Riemannian curvature; scalar curvature
UR - http://eudml.org/doc/250504
ER -

References

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  1. Abbassi, M. T. K., Calvaruso, G., 10.1007/s00605-006-0421-9, Monatsh. Math. 151 (2) (2007), 89–109. (2007) Zbl1128.53049MR2322938DOI10.1007/s00605-006-0421-9
  2. Adams, J. F., 10.2307/1970147, Ann. Math. 72 (1960), 20–104. (1960) Zbl0096.17404MR0141119DOI10.2307/1970147
  3. Besse, A. L., Einstein Manifolds, Springer-Verlag, Berlin–Heidelberg–New York, 1987. (1987) Zbl0613.53001MR0867684
  4. Blair, D., When is the tangent sphere bundle locally symmetric?, Geom. Topol., World Sci. Publishing, Singapore (1989), 15–30. (1989) MR1001586
  5. Boeckx, E., Vanhecke, L., Characteristic reflections on unit tangent sphere bundles, Houston J. Math. 23 (1997), 427–448. (1997) Zbl0897.53010MR1690045
  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. (1997) 
  7. Boeckx, E., Vanhecke, L., Curvature homogeneous unit tangent sphere bundles, Publ. Math. Debrecen 35 (1998), 389–413. (1998) MR1657491
  8. Boeckx, E., Vanhecke, L., 10.1023/A:1004629423529, Period. Math. Hungar. 36 (1998), 79–95. (1998) MR1694613DOI10.1023/A:1004629423529
  9. Boeckx, E., Vanhecke, L., 10.1016/S0926-2245(00)00021-8, Differential Geom. Appl. 13 (2000), 77–93. (2000) Zbl0973.53053MR1775222DOI10.1016/S0926-2245(00)00021-8
  10. Boeckx, E., Vanhecke, L., 10.1023/A:1013779805244, Czechoslovak Math. J. 51 (126) (2001), 523–544. (2001) Zbl1079.53063MR1851545DOI10.1023/A:1013779805244
  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. (1987) 
  12. Borisenko, A. A., Yampolsky, A. L., The sectional curvature of the Sasaki metric of T r M n , Ukrain. Geom. Sb. 30 (1987), 10–17. (1987) 
  13. Borisenko, A. A., Yampolsky, A. L., 10.1070/RM1991v046n06ABEH002859, Russian Math. Surveys 46 (6) (1991), 55–106. (1991) MR1164201DOI10.1070/RM1991v046n06ABEH002859
  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. (2005) Zbl1079.53045MR2105140
  15. Ivanov, S., Petrova, I., 10.1023/A:1005014507809, Geom. Dedicata 70 (1998), 269–282. (1998) Zbl0903.53016MR1624814DOI10.1023/A:1005014507809
  16. Kobayashi, S., Nomizu, K., Foundations of Differential Geometry II, Interscience Publishers, New York–London–Sydney, 1969. (1969) MR0238225
  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. (2000) Zbl1024.53030MR1848571
  18. Kowalski, O., Sekizawa, M., 10.1023/A:1006707521207, Ann. Global Anal. Geom. 18 (2000), 207–219. (2000) Zbl1011.53025MR1795094DOI10.1023/A:1006707521207
  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. (2000) Zbl1163.53321MR1848571
  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. (2002) Zbl1039.53050MR1962729
  21. Kowalski, O., Sekizawa, M., Vlášek, Z., 10.1090/conm/288/04820, Global Differential Geometry: The Mathematical Legacy of Alfred Gray (Fernandez, M. and Wolf, J. A., eds.), Contemp. Math. 288 (2001), 110–118. (2001) Zbl1011.53034MR1871003DOI10.1090/conm/288/04820
  22. Nagy, P. T., Geodesics on the tangent sphere bundle of a Riemannian manifold, Geom. Dedicata 7 (1978), 233–243. (1978) Zbl0385.53010MR0487892
  23. Nash, J., Positive Ricci curvature on fiber bundles, J. Differential Geom. 14 (1979), 241–254. (1979) MR0587552
  24. Podestà, F., Isometries of tangent sphere bundles, Boll. Un. Mat. Ital. A(7) 5 (1991), 207–214. (1991) MR1120381
  25. Poor, W., 10.1007/BF01430964, Math. Ann. 216 (1975), 245–252. (1975) Zbl0293.53016MR0400110DOI10.1007/BF01430964
  26. Takagi, H., 10.2748/tmj/1178241040, Tôhoku Math. J. 27 (1975), 103–110. (1975) Zbl0323.53037MR0442852DOI10.2748/tmj/1178241040
  27. Wolf, J. A., Elliptic spaces in Grassmann manifolds, Illinois J. Math. 7 (1963), 447–462. (1963) MR0156295
  28. Yampolsky, A. L., On the geometry of tangent sphere bundles of Riemannian manifolds, Ukrain. Geom. Sb 24 (1981), 129–132, in Russian. (1981) MR0629822
  29. Yampolsky, A. L., On Sasaki metric of tangent and normal bundle, Ph.D. thesis, Odessa, 1986, (Russian). (1986) 

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