Keywords: Riemannian metric; tangent bundle; tangent sphere bundle; Riemannian curvature; scalar curvature
@article{ARM_2008_44_5_a5,
author = {Kowalski, Old\v{r}ich and Sekizawa, Masami},
title = {On {Riemannian} geometry of tangent sphere bundles with arbitrary constant radius},
journal = {Archivum mathematicum},
pages = {391--401},
year = {2008},
volume = {44},
number = {5},
mrnumber = {2501575},
zbl = {1212.53043},
language = {en},
url = {http://geodesic.mathdoc.fr/item/ARM_2008_44_5_a5/}
}
Kowalski, Oldřich; Sekizawa, Masami. On Riemannian geometry of tangent sphere bundles with arbitrary constant radius. Archivum mathematicum, Tome 44 (2008) no. 5, pp. 391-401. http://geodesic.mathdoc.fr/item/ARM_2008_44_5_a5/
[1] Abbassi, M. T. K., Calvaruso, G.: $g$-natural contact metrics on unit tangent sphere bundles. Monatsh. Math. 151 (2) (2007), 89–109. | DOI | MR | Zbl
[2] Adams, J. F.: On the non-existence of elements of Hopf invariant one. Ann. Math. 72 (1960), 20–104. | DOI | MR | Zbl
[3] Besse, A. L.: Einstein Manifolds. Springer-Verlag, Berlin–Heidelberg–New York, 1987. | MR | Zbl
[4] Blair, D.: When is the tangent sphere bundle locally symmetric?. Geom. Topol., World Sci. Publishing, Singapore (1989), 15–30. | MR
[5] Boeckx, E., Vanhecke, L.: Characteristic reflections on unit tangent sphere bundles. Houston J. Math. 23 (1997), 427–448. | MR | Zbl
[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
[8] Boeckx, E., Vanhecke, L.: Unit tangent sphere bundles and two-point homogeneous spaces. Period. Math. Hungar. 36 (1998), 79–95. | DOI | MR
[9] Boeckx, E., Vanhecke, L.: Harmonic and minimal vector fields on tangent and unit tangent bundles. Differential Geom. Appl. 13 (2000), 77–93. | DOI | MR | Zbl
[10] Boeckx, E., Vanhecke, L.: Unit tangent sphere bundles with constant scalar curvature. Czechoslovak Math. J. 51 (126) (2001), 523–544. | DOI | MR | Zbl
[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 | MR
[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 | Zbl
[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 | MR | Zbl
[16] Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry II. Interscience Publishers, New York–London–Sydney, 1969. | MR
[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 | Zbl
[18] Kowalski, O., Sekizawa, M.: On tangent sphere bundles with small or large constant radius. Ann. Global Anal. Geom. 18 (2000), 207–219. | DOI | MR | Zbl
[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 | Zbl
[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 | Zbl
[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 | MR | Zbl
[22] Nagy, P. T.: Geodesics on the tangent sphere bundle of a Riemannian manifold. Geom. Dedicata 7 (1978), 233–243. | MR | Zbl
[23] Nash, J.: Positive Ricci curvature on fiber bundles. J. Differential Geom. 14 (1979), 241–254. | MR
[24] Podestà, F.: Isometries of tangent sphere bundles. Boll. Un. Mat. Ital. A(7) 5 (1991), 207–214. | MR
[25] Poor, W.: Some exotic spheres with positive Ricci curvature. Math. Ann. 216 (1975), 245–252. | DOI | MR | Zbl
[26] Takagi, H.: Conformally flat Riemannian manifolds admitting a transitive group of isometries. Tôhoku Math. J. 27 (1975), 103–110. | DOI | MR | Zbl
[27] Wolf, J. A.: Elliptic spaces in Grassmann manifolds. Illinois J. Math. 7 (1963), 447–462. | MR
[28] Yampolsky, A. L.: On the geometry of tangent sphere bundles of Riemannian manifolds. Ukrain. Geom. Sb 24 (1981), 129–132, in Russian. | MR
[29] Yampolsky, A. L.: On Sasaki metric of tangent and normal bundle. Ph.D. thesis, Odessa, 1986, (Russian).