Previous |  Up |  Next

Article

Keywords:
Riemannian metric; tangent bundle; tangent sphere bundle; Riemannian curvature; scalar curvature
Summary:
We shall survey our work on Riemannian geometry of tangent sphere bundles with arbitrary constant radius done since the year 2000.
References:
[1] Abbassi, M. T. K., Calvaruso, G.: $g$-natural contact metrics on unit tangent sphere bundles. Monatsh. Math. 151 (2) (2007), 89–109. DOI 10.1007/s00605-006-0421-9 | MR 2322938 | Zbl 1128.53049
[2] Adams, J. F.: On the non-existence of elements of Hopf invariant one. Ann. Math. 72 (1960), 20–104. DOI 10.2307/1970147 | MR 0141119 | Zbl 0096.17404
[3] Besse, A. L.: Einstein Manifolds. Springer-Verlag, Berlin–Heidelberg–New York, 1987. MR 0867684 | Zbl 0613.53001
[4] Blair, D.: When is the tangent sphere bundle locally symmetric?. Geom. Topol., World Sci. Publishing, Singapore (1989), 15–30. MR 1001586
[5] Boeckx, E., Vanhecke, L.: Characteristic reflections on unit tangent sphere bundles. Houston J. Math. 23 (1997), 427–448. MR 1690045 | Zbl 0897.53010
[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.
[7] Boeckx, E., Vanhecke, L.: Curvature homogeneous unit tangent sphere bundles. Publ. Math. Debrecen 35 (1998), 389–413. MR 1657491
[8] Boeckx, E., Vanhecke, L.: Unit tangent sphere bundles and two-point homogeneous spaces. Period. Math. Hungar. 36 (1998), 79–95. DOI 10.1023/A:1004629423529 | MR 1694613
[9] Boeckx, E., Vanhecke, L.: Harmonic and minimal vector fields on tangent and unit tangent bundles. Differential Geom. Appl. 13 (2000), 77–93. DOI 10.1016/S0926-2245(00)00021-8 | MR 1775222 | Zbl 0973.53053
[10] Boeckx, E., Vanhecke, L.: Unit tangent sphere bundles with constant scalar curvature. Czechoslovak Math. J. 51 (126) (2001), 523–544. DOI 10.1023/A:1013779805244 | MR 1851545 | Zbl 1079.53063
[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.
[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.
[13] Borisenko, A. A., Yampolsky, A. L.: Riemannian geometry of fiber bundles. Russian Math. Surveys 46 (6) (1991), 55–106. DOI 10.1070/RM1991v046n06ABEH002859 | MR 1164201
[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. MR 2105140 | Zbl 1079.53045
[15] Ivanov, S., Petrova, I.: Riemannian manifold in which the skew-symmetric curvature operator has pointwise constant eigenvalues. Geom. Dedicata 70 (1998), 269–282. DOI 10.1023/A:1005014507809 | MR 1624814 | Zbl 0903.53016
[16] Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry II. Interscience Publishers, New York–London–Sydney, 1969. MR 0238225
[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. MR 1848571 | Zbl 1024.53030
[18] Kowalski, O., Sekizawa, M.: On tangent sphere bundles with small or large constant radius. Ann. Global Anal. Geom. 18 (2000), 207–219. DOI 10.1023/A:1006707521207 | MR 1795094 | Zbl 1011.53025
[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. MR 1848571 | Zbl 1163.53321
[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. MR 1962729 | Zbl 1039.53050
[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. DOI 10.1090/conm/288/04820 | MR 1871003 | Zbl 1011.53034
[22] Nagy, P. T.: Geodesics on the tangent sphere bundle of a Riemannian manifold. Geom. Dedicata 7 (1978), 233–243. MR 0487892 | Zbl 0385.53010
[23] Nash, J.: Positive Ricci curvature on fiber bundles. J. Differential Geom. 14 (1979), 241–254. MR 0587552
[24] Podestà, F.: Isometries of tangent sphere bundles. Boll. Un. Mat. Ital. A(7) 5 (1991), 207–214. MR 1120381
[25] Poor, W.: Some exotic spheres with positive Ricci curvature. Math. Ann. 216 (1975), 245–252. DOI 10.1007/BF01430964 | MR 0400110 | Zbl 0293.53016
[26] Takagi, H.: Conformally flat Riemannian manifolds admitting a transitive group of isometries. Tôhoku Math. J. 27 (1975), 103–110. DOI 10.2748/tmj/1178241040 | MR 0442852 | Zbl 0323.53037
[27] Wolf, J. A.: Elliptic spaces in Grassmann manifolds. Illinois J. Math. 7 (1963), 447–462. MR 0156295
[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
[29] Yampolsky, A. L.: On Sasaki metric of tangent and normal bundle. Ph.D. thesis, Odessa, 1986, (Russian).
Partner of
EuDML logo