# PSEUDO-SYMMETRY ON UNIT TANGENT SPHERE BUNDLES

• Cho, Jong Taek (Department of Mathematics, Chonnam National University) ;
• Chun, Sun Hyang (Department of Mathematics, Chosun University)
• Accepted : 2016.03.15
• Published : 2016.06.25
• 77 11

#### Abstract

In this paper, we study the pseudo-symmetry of unit tangent sphere bundle. We prove that if the unit tangent sphere bundle $T_1M$ with standard contact metric structure over a locally symmetric $M^n$, $n{\geq}3$ is pseudo-symmetric, then M is of constant curvature.

#### Keywords

pseudo-symmetry;unit tangent sphere bundles

#### Acknowledgement

Supported by : Chosun University

#### References

1. D. E. Blair, Riemannian geometry of contact and symplectic manifolds, Second edition, Progr. Math. 203, Birkhauser Boston, Inc., Boston, MA, 2010.
2. D. E. Blair, When is the tangent sphere bundle locally symmetric?, Geometry and Topology, World Scientific, Singapore 509 (1989), 15-30.
3. E. Boeckx and L.Vanhecke, Characteristic reflections on unit tangent sphere bundles, Houston J. Math., 23 (1997), 427-448.
4. E. Boeckx, D.Perrone and L.Vanhecke, Unit tangent sphere bundles and two-point homogeneous spaces, Periodica Math. Hungarica, 36 (1998), 79-95. https://doi.org/10.1023/A:1004629423529
5. E. Boeckx and G. Calvaruso, When is the unit tangent sphere bundle semi-symmetric?, Tohoku Math. J., 56 (2004), 357-366. https://doi.org/10.2748/tmj/1113246672
6. E. Cartan, Lecons sur la geometrie des espaces de Riemann, Gauthier-Villars, Paris, 1946.
7. J.T. Cho and S.H. Chun, Symmetries on unit tangent sphere bundles, roceedings of The Eleven InternationalWorkshop on Differential Geom., 11 (2007), 153-170.
8. J.T. Cho and Jun-ichi Inoguchi, Pseudo-symmetric contact 3-manifolds II, Note di Matematica, 27 (2007), 119-129.
9. R. Deszcz, On pseudosymmetric spaces, Bull. Soc. Math. Belg., 44 (1992), 1-34.
10. P. Dombrowski, On the geometry of the tangent bundle, J. Reine Angew. Math., 210 (1962), 73-88.
11. O. Kowalski, Curvature of the induced Riemannian metric of the tangent bundle of a Riemannian manifold, J. Reine Angew. Math., 250 (1971), 124-129.
12. Y. Tashiro, On contact structures of unit tangent sphere bundles, Tohoku Math. J., 21 (1969), 117-143. https://doi.org/10.2748/tmj/1178243040
13. K. Yano and S. Ishihara, Tangent and cotangent bundles, M. Dekker Inc., 1973.