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INVARIANT SUBMANIFOLDS OF CONTACT (κ, μ)-MANIFOLDS

Published online by Cambridge University Press:  01 September 2008

BENIAMINO CAPPELLETTI MONTANO ,
LUIGIA DI TERLIZZI  and
MUKUT MANI TRIPATHI
BENIAMINO CAPPELLETTI MONTANO
Affiliation:
Dipartimento di Matematica, Università degli Studi di Bari, Via E. Orabona 4, 70125 Bari, Italy e-mail: [email protected]
LUIGIA DI TERLIZZI
Affiliation:
Dipartimento di Matematica, Università degli Studi di Bari, Via E. Orabona 4, 70125 Bari, Italy e-mail: [email protected]
MUKUT MANI TRIPATHI
Affiliation:
Department of Mathematics and Astronomy, Lucknow University, Lucknow 226 007, India
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Abstract

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Invariant submanifolds of contact (κ, μ)-manifolds are studied. Our main result is that any invariant submanifold of a non-Sasakian contact (κ, μ)-manifold is always totally geodesic and, conversely, every totally geodesic submanifold of a non-Sasakian contact (κ, μ)-manifold, μ ≠ 0, such that the characteristic vector field is tangent to the submanifold is invariant. Some consequences of these results are then discussed.

Keywords

Primary 53C40Secondary 53C2553D1053D15
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 50 , Issue 3 , September 2008 , pp. 499 - 507
Copyright
Copyright © Glasgow Mathematical Journal Trust 2008

References

REFERENCES

1.Agut, C., Submanifolds in contact metric manifolds with a nullity condition, An. Univ. Timişoara Ser. Mat.-Inform. 41 (2003), 3–9.Google Scholar
2.Blair, D. E., Two remarks on contact metric structures, Tôhoku Math. J. 29 (1977), 319–324.Google Scholar
3.Blair, D. E., Riemannian geometry of contact and symplectic manifolds, Progress in Mathematics, 203. (Birkhäuser Boston, Inc., Boston, MA, 2002).Google Scholar
4.Blair, D. E., Koufogiorgos, T. and Papantoniou, B. J., Contact metric manifolds satisfyng a nullity condition, Israel J. Math. 91 (1995), 189–214.Google Scholar
5.Boeckx, E., A class of locally φ-symmetric contact metric spaces, Arch. Math. (Basel) 72 (1999), 466–472.CrossRefGoogle Scholar
6.Boeckx, E., Contact-homogeneous locally φ-symmetric manifolds, Glasg. Math. J. 48 (2006), 93–109.Google Scholar
7.Boeckx, E., A full classification of contact metric (κ, μ)-spaces, Illinois J. Math. 44 (2000), 212–219.CrossRefGoogle Scholar
8.Cappelletti, B.Montano, Bi-Legendrian connections, Ann. Polon. Math. 86 (2005), 79–95.Google Scholar
9.Cappelletti, B. Montano, Some remarks on the generalized Tanaka–Webster connection of a contact metric manifold, Rocky Mountain J. Math., to appear.Google Scholar
10.Cappelletti Montano, B. and Di Terlizzi, L., Contact metric (κ, μ)-spaces as bi-Legendrian manifolds, Bull. Austral. Math. Soc., to appear.Google Scholar
11.Chinea, D., Invariant submanifolds of a quasi-K-Sasakian manifold, Riv. Mat. Univ. Parma 11 (1985), 25–29.Google Scholar
12.Dombrowski, P., On the geometry of the tangent bundle, J. Rein Angew. Math. 210 (1962), 73–88.Google Scholar
13.Endo, H., Invariant submanifolds in a contact Riemannian manifold, Tensor 42 (1985), 86–89.Google Scholar
14.Endo, H., On the curvature tensor field of a type of contact metric manifolds and of its certain submanifolds, Publ. Math. Debrecen 48 (1996), 253–269.CrossRefGoogle Scholar
15.Kon, M., Invariant submanifolds of normal contact metric manifolds, Kōdai Math. Sem. Rep. 27 (1973), 330–336.Google Scholar
16.Koufogiorgos, T., Contact Riemannian manifolds with constant φ-sectional curvature, Tokyo J. Math. 20 (1997), 13–22.Google Scholar
17.Kowalski, O., Curvature of the induced Riemannian metric on the tangent bundle, J. Rein Angew. Math. 250 (1971), 124–129.Google Scholar
18.Murathan, C., Arslan, K., Özgür, C. and Yildiz, A., On invariant submanifolds satisfying a nullity condition, An. Univ. Bucureşti Mat. Inform. 50 (2001), 103–113.Google Scholar
19.Sasaki, S., On the differential geometry of tangent bundle of Riemannian manifolds, Tôhoku Math. J. 10 (1958), 338–354.CrossRefGoogle Scholar
20.Sasaki, S., On the differential manifolds with contact metric structures. Part II, Tôhoku Math. J. 14 (1962), 146–155.Google Scholar
21.Sasaki, S. and Hatakeyama, Y., On differentiable geometry of tangent bundle of Riemannian manifolds, J. Math. Soc. Japan 14 (1962), 249–271.Google Scholar
22.Tashiro, Y., On contact structures of tangent sphere bundles, Tôhoku Math. J. 21 (1969), 117–143.Google Scholar
23.Tripathi, M. M., Sasahara, T. and Kim, J.-S., On invariant submanifolds of contact metric manifolds, Tsukuba J. Math. 29 (2) (2005), 495–510.Google Scholar
24.Yano, K. and Kon, M., Structures on manifolds, Series in Pure Mathematics (World Scientific, Singapore, 1984).Google Scholar