Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-25T17:28:04.214Z Has data issue: false hasContentIssue false
  • English
  • Français

On the non-regularity of tangent sphere bundles

Published online by Cambridge University Press:  14 November 2011

David E. Blair
David E. Blair
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, Michigan, U.S.A..
Get access Rights & Permissions [Opens in a new window]

Synopsis

Classically the tangent sphere bundles have formed a large class of contact manifolds; their contact structures are not in general regular, however. Specifically we prove that the natural contact structure on the tangent sphere bundle of a compact Riemannian manifold of non-positive constant curvature is not regular.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 82 , Issue 1-2 , 1978 , pp. 13 - 17
Copyright
Copyright © Royal Society of Edinburgh 1978

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Blair, D. E.. On the non-existence of flat contact metric structures. Tohoku Math. J. 28 (1976), 373–379.Google Scholar
2Blair, D. E.. Contact Manifolds in Riemannian Geometry. Lecture Notes in Mathematics 509 (Berlin: Springer, 1976).Google Scholar
3Boothby, W. M. and Wang, H. C.. On contact manifolds. Ann. of Math. 68 (1958), 721–734.Google Scholar
4Dombrowski, P.. On the geometry of the tangent bundle. J. Reine Angew. Math. 210 (1962), 73–88.CrossRefGoogle Scholar
5Sasaki, S.. On the differential geometry of tangent bundles of Riemannian manifolds. Tohoku Math. J. 10 (1958), 338–354.CrossRefGoogle Scholar
6Sasaki, S.. On differentiable manifolds with certain structures which are closely related to almost contact structure I. Tohoku Math. J. 12 (1960), 459–476.Google Scholar
7Steenrod, N.. Topology of Fibre Bundles. (Princeton Univ. Press, 1951).Google Scholar
8Tachibana, S. and Okumura, M.. On the almost complex structure of tangent bundles of Riemannian spaces. Tohoku Math. J. 14 (1962), 156–161.Google Scholar
9Tanno, S.. Sasakian manifolds with constant φ-holomorphic sectional curvature. Tôhoku Math. J. 21 (1969), 501–507.Google Scholar
10Tashiro, Y.. On contact structures of tangent sphere bundles. Tôhoku Math. J. 21 (1969), 117–143.Google Scholar
11Yano, K. and Ishihara, S.. Tangent and Cotangent Bundles. (New York: Dekker, 1973).Google Scholar