Research ArticleNo Access

Some characterizations of anti-invariant submanifolds of trans-sasakian manifolds

A. Sarkar

https://orcid.org/0000-0002-7370-1698

Department of Mathematics, University of Kalyani, Kalyani 741235, West Bengal, India

E-mail Address: avjaj@yahoo.co.in

Search for more papers by this author

Pradip Bhakta

Department of Mathematics, University of Kalyani, Kalyani 741235, West Bengal, India

E-mail Address: pradip020791@gmail.com

Search for more papers by this author

, and

Matilal Sen

Department of Mathematics, Saldiha College Saldiha, Bankura 722173, West Bengal, India

E-mail Address: matilal_sen@yahoo.co.in

Search for more papers by this author

https://doi.org/10.1142/S1793557121501679Cited by:1 (Source: Crossref)

Abstract

The object of this paper is to study anti-invariant submanifolds of trans-Sasakian manifolds. We characterize such submanifolds on the basis of parallelism, semi-parallelism and pseudo parallelism of the second fundamental form of the submanifolds. We also characterize totally umbilical anti-invariant submanifolds of trans-Sasakian manifolds. Existence of Legendre curves on such submanifolds has been analyzed. Whether an anti-invariant submanifold of a trans-Sasakian manifold inherits local symmetry from ambient space is investigated here. Nature of Ricci soliton on anti-invariant submanifolds of trans-Sasakian manifolds has been characterized.

Communicated by K. Saito

Keywords:

AMSC: 53C40, 53C25

References

1. C. Baikoussis and D. E. Blair, On Legendre curves in contact 3-manifolds, Geom. Dedicata 49 (1994) 135–142. Crossref, Web of Science, Google Scholar
2. J. K. Beem, P. E. Ehrlich and T. G. Powell, Warped product manifolds, Selected studies, in Physics-Astrophysics-Mathematics, History of Science (North Holland, New York, 1982). Google Scholar
3. D. E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds (Birkh $ä$ user, Boston–Basel–Berlin, 2002). Crossref, Google Scholar
4. D. E. Blair and J. A. Oubina, Conformal and related changes of metric on the product of two almost contact metric manifolds, Publ. Math. 34 (1990) 199–207. Crossref, Google Scholar
5. B. Y. Chen, Geometry of Submanifolds (Marcel Dekker, New York, 1973). Google Scholar
6. B. Y. Chen, Totally umbilical submanifolds, Soochow J. Math. 5 (1979) 9–36. Google Scholar
7. B. Y. Chen, Geometry of warped products as Riemannian submanifolds and related problems, Soochow J. Math. 28 (2002) 125–156. Google Scholar
8. D. Chinea and C. Gonzalez, Curvature relations in trans-Sasakian manifolds, in Proc. XIIth Portuguese–Spanish Conf. Mathematics, Vol. II (Portuguese), Braga (University of Minho, Braga, 1987), pp. 564–571. Google Scholar
9. D. Chinea and P. S. Prestelo, Invariant submanifolds of a trans-Sasakian manifold, Publ. Math. Debreceen 38 (1991) 103–109. Crossref, Web of Science, Google Scholar
10. A. De, Totally geodesic submanifolds of a trans-Sasakian manifold, Proc. Estonian Acad. Sci. 62 (2013) 249–257. Crossref, Web of Science, Google Scholar
11. U. C. De and A. Sarkar, On three-dimensional trans-Sasakian manifolds, Extracta Math. 23 (2008) 265–277. Google Scholar
12. U. C. De and A. Sarkar, On pseudoslant submanifolds of trans-Sasakian manifolds, Proc. Estonian Acad. Sci. 60 (2011) 1–15. Crossref, Web of Science, Google Scholar
13. U. C. De and M. M. Tripathi, Ricci tensor in 3-dimensional trans-Sasakian manifolds, Kyungpook Math. J. 43 (2003) 247–255. Google Scholar
14. S. Deshmukh and M. A. Al-Gwaiz, Totally umbilical submanifolds of a Sasakian manifold, New Zealand J. Math. 22 (1993) 43–47. Google Scholar
15. R. Deszcz, On pseudo symmetric spaces, Bull. Soc. Math. Belg. Ser. A. 44 (1992) 1–34. Web of Science, Google Scholar
16. T. Friedrich and S. Ivanov, Almost contact manifolds with torsion and parallel spinors, J. Reine Angew. Math. 559 (2003) 217–236. Crossref, Web of Science, Google Scholar
17. Z. Guojing and W. Jianguo, Invariant submanifolds and modes of non-linear autonomous systems, Appl. Math. Mech. 19 (1998) 587–693. Crossref, Google Scholar
18. A. Gray and L. M. Hervella, The sixteen classes of almost Harmite manifolds and their linear invariants, Ann. Mat., Pura Appl. 123(4) (1980) 35–58. Crossref, Google Scholar
19. R. S. Hamilton, The Ricci flow on surfaces, in Mathematics and General Relativity, Contemporary Mathematics, Vol. 71 (AMS, 1988). Crossref, Google Scholar
20. S. K. Hui, S. Uddin and P. Mandal, Submanifolds of generalized $(κ, μ)$ -space form, Period. Math. Hungar. 77 (2018) 329–339. Crossref, Web of Science, Google Scholar
21. K. Kenmotsu, A class of almost contact Riemannian manifolds, Töhoku Math. J. 24 (1972) 93–100. Crossref, Google Scholar
22. M. Kobayashi, Semi-invariant submanifolds of a certain class of almost contact metric manifolds, Tensor 43 (1986) 28–36. Google Scholar
23. M. Kon, Invariant submanifolds of normal contact metric manifolds, Kodai Math. Sem. Rep. 27 (1973) 330–336. Google Scholar
24. O. A. Ladyzhenskaya, Boundary Value Problem in Mathematical Physics, Vol. 13 (AMS) (1989). Google Scholar
25. J. C. Marrero, The local structure of trans-Sasakian manifolds, Ann. Mat. Pura Appl. 162 (1992) 77–86. Crossref, Web of Science, Google Scholar
26. B. O’Neill, Manifolds of negetive curvature, Trans. Am. Math. Soc. 145 (1969) 1–49. Crossref, Web of Science, Google Scholar
27. O’Neill B., Semi-Riemannian Geometry with Applications to Relativity, Pure and Applied Mathematics, Vol. 103 (Academic Press, New York, 1983). Google Scholar
28. J. A. Oubina, New classes of almost contact metric structures, Publ. Math. Debrecen 32 (1985) 187–193. Crossref, Web of Science, Google Scholar
29. G. Perelman, The entropy formula for the Ricci flow and its geometric applications, preprint (2010), arxiv:1021111599. Google Scholar
30. R. Prasad, Quasi-conformal curvature on trans-Sasakian manifolds, Proc. Indian Nat. Sci. Acad. 76 (2010) 7–35. Google Scholar
31. W. Roter, On conformally recurrent, Ricci recurrent manifolds, Colloq. Math. 46 (1982) 45–57. Crossref, Google Scholar
32. M. H. Sahid, Some results on anti invariant submanifolds of a trans-Sasakian manifold, Bull. Malayas. Math. Sci. Soc. 27 (2004) 117–127. Google Scholar
33. A. Sarkar, Submanifolds of Sasakian manifolds, Lobachevskii J. Math. 32 (2011) 87–93. Crossref, Google Scholar
34. A. Sarkar and N. Biswas, Some conditions for totally umbilical submanifolds of trans-Sasakian manifolds to be totally geodesic, to appear in Acta Math. Univ. Com. (2020). Google Scholar
35. A. Sarkar and S. K. Hui and M. Sen, A study on Legendre curves in 3-dimensional trans-Sasakian manifolds, Lobachevskii J. Math. 35 (2014) 11–18. Crossref, Google Scholar
36. A. Sarkar and M. Sen, On invariant submanifolds of trans-Sasakian manifolds, Proc. Estonian Acad. Sci. 61 (2012) 29–37. Crossref, Web of Science, Google Scholar
37. A. Sarkar and M. Sen, Invariant submanifolds of LP-Sasakian manifolds, Extracta Math. 27 (2012) 145–154. Google Scholar
38. A. Sarkar, A. Sil and A. Paul, Some characterizations of $η$ -Ricci solitons and Kagan sub projective spaces, Ukrainian Math. J. 72 (2020) 488–494. Crossref, Web of Science, Google Scholar
39. S. Sasaki and Y. Hatakeyama, On differentiable manifolds with contact metric structure, J. Math. Soc. Japan 14 (1962) 249–271. Crossref, Google Scholar
40. R. Sharma, Certain results on $K$ -Contact and $(κ, μ)$ -contact manifolds, J. Geom. 89 (2008) 138–247. Crossref, Google Scholar
41. S. Sular and C. Ȯzgur, On some submanifolds of Kenmotsu manifolds, Chaos, Solitons and Fractals 42 (2009) 1990–1995. Crossref, Web of Science, Google Scholar
42. T. Takahashi, Sasakian $ϕ$ -symmetric spaces, Tohoku Math. J. 29 (1977) 91–113. Crossref, Google Scholar
43. S. Tanno, The automorphism groups of almost contact Riemannian manifolds, Tohoku Math. J. 2 (1969) 21–38. Google Scholar
44. J. Welyczko, On Legendre curves in 3-dimensional normal almost contact metric manifolds, Soochow J. Math. 33 (2007) 929–937. Google Scholar
45. K. Yano and M. Kon, Anti invariant submanifolds of Sasakian space forms II, J. Korean Math. Soc. 13 (1976) 1–14. Google Scholar
46. K. Yano and M. Kon, Anti Invariant Submanifolds, Pure and Applied Mathematics, Vol. 21 (Marcel Dekker, New York, 1976). Google Scholar
47. K. Yano and M. Kon, Anti invariant submanifolds of Sasakian space forms, Tohoku Math. J. 9 (1977) 9–23. Google Scholar
48. K. Yano and M. Kon, Structures on Manifolds, Series in Pure Mathematics (World Scientific Publishing, Springer, 1984). Link, Google Scholar