Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



When are the tangent sphere bundles of a Riemannian manifold reducible?

Author: E. Boeckx
Journal: Trans. Amer. Math. Soc. 355 (2003), 2885-2903
MSC (2000): Primary 53B20, 53C12, 53C20
Published electronically: March 14, 2003
MathSciNet review: 1975404
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We determine all Riemannian manifolds for which the tangent sphere bundles, equipped with the Sasaki metric, are local or global Riemannian product manifolds.

References [Enhancements On Off] (What's this?)

  • 1. M. F. Atiyah, R. Bott and A. Shapiro, Clifford modules, Topology 3, Suppl. 1 (1964), 3-38. MR 29:5250
  • 2. J. Berndt, E. Boeckx, P. Nagy and L. Vanhecke, Geodesics on the unit tangent bundle, preprint, 2001.
  • 3. J. Berndt, F. Tricerri and L. Vanhecke, Generalized Heisenberg groups and Damek-Ricci harmonic spaces, Lecture Notes in Math. 1598, Springer-Verlag, Berlin, Heidelberg, New York, 1995. MR 97a:53068
  • 4. E. Boeckx and G. Calvaruso, When is the unit tangent sphere bundle semi-symmetric?, preprint, 2002.
  • 5. E. Boeckx and L. Vanhecke, Characteristic reflections on unit tangent sphere bundles, Houston J. Math. 23 (1997), 427-448. MR 2000e:53052
  • 6. E. Boeckx and L. Vanhecke, Curvature homogeneous unit tangent sphere bundles, Publ. Math. Debrecen 53 (1998), 389-413. MR 2000d:53080
  • 7. E. Boeckx and L. Vanhecke, Harmonic and minimal vector fields on tangent and unit tangent bundles, Differential Geom. Appl. 13 (2000), 77-93. MR 2001f:53138
  • 8. G. de Rham, Sur la reductibilité d'un espace de Riemann, Comment. Math. Helv. 26 (1952), 328-344. MR 14:584a
  • 9. O. Kowalski and M. Sekizawa, On tangent sphere bundles with small or large constant radius, Ann. Global Anal. Geom. 18 (2000), 207-219. MR 2001i:53049
  • 10. O. Kowalski, M. Sekizawa and Z. Vlásek, Can tangent sphere bundles over Riemannian manifolds have strictly positive sectional curvature?, in: Global Differential Geometry: The Mathematical Legacy of Alfred Gray (eds. M. Fernández, J. A. Wolf), Contemp. Math. 288, Amer. Math. Soc., Providence, RI, 2001, 110-118. MR 2002i:53047

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 53B20, 53C12, 53C20

Retrieve articles in all journals with MSC (2000): 53B20, 53C12, 53C20

Additional Information

E. Boeckx
Affiliation: Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium

Keywords: Tangent sphere bundle, Sasaki metric, reducibility, Clifford structures, foliations
Received by editor(s): November 11, 2002
Received by editor(s) in revised form: January 21, 2003
Published electronically: March 14, 2003
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society