On Some Classes of Invariant Submanifolds of \((k,\mu)\)-Contact Manifolds

Authors

  • M. S. Siddesha
  • C. S. Bagewadi Emeritus Fellow, Department of Mathematics, Kuvempu University, Jnana Sahyadri, Shankaraghatta 577 451, Karnataka

DOI:

https://doi.org/10.26713/jims.v9i1.451

Keywords:

Invariant submanifold, \((k, \mu)\)-contact manifold, Totally geodesic, Pseudoparallel, Ricci-generalized pseudoparallel

Abstract

In this paper, we study invariant submanifolds of \((k,\mu)\)-contact manifolds. We consider pseudoparallel, 2-pseudoparallel, Ricci-generalized pseudoparallel, 2-Ricci-generalized pseudoparallel submanifolds of \((k,\mu)\)-contact manifolds. Further, we search for the conditions \(\mathcal{Z}(X,Y)\cdot\sigma = 0\) and \(\mathcal{Z}(X,Y )\cdot\bar{\nabla}\sigma =0\) on invariant submanifolds of \((k,\mu)\)-contact manifolds, where $\mathcal{Z}$ is the concircular curvature tensor.

Downloads

Download data is not yet available.

References

B.S. Anitha and C.S. Bagewadi, Invariant submanifolds of Sasakian manifolds, Proceedings of the National Conference on Differential Geometry, (2013) 28-36.

B.S. Anitha and C.S. Bagewadi, Invariant Submanifolds of Sasakian Manifolds Admitting Semisymmetric Metric Connection, Communications in Mathematics and Applications Volume 4 (1) (2013), 2938.

A.C. Asperti, G.A. Lobos and F. Mercuri, Pseudo-parallel immersions in space forms, Math. Contemp. 17 (1999), 59-70.

A.C. Asperti, G.A. Lobos and F. Mercuri, pseudoparallel immersions of a space form, Adv. Geom. 2 (2002), 57-71.

Avik De, A note on invariant submanifolds of (k, μ)-contact manifolds, Ukranian Mathematical J. 62(11) (2011), 1803-1809.

D.E. Blair, Contact manifolds in Riemannian geometry, Lecture notes in Math., 509, Springer-Verlag, Berlin, (1976).

D.E. Blair, Two remarks on contact metric structures, Tohoku Math. J., 29 (1977), 319-324.

D.E. Blair, T. Koufogiorgos and B.J. Papantoniou, Contact metric manifolds satisfyng a nullity condition, Israel J. Math., 91 (1995), 189-214.

E. Boeckx, A full classification of contact metric (k, μ)-spaces, Illinois J. Math. 44 (2000), 212-219.

B.Y. Chen, Geometry of submanifolds and its applications, Science University of Tokyo, Tokyo, (1981).

F. Defever, J.M. Morvan, I. Van de Woestijne, L. Verstaelen and G. Zafindratafa, Geometry and topology of submanifolds IX, World Scientific Publishing (1999).

J. Deprez, Semiparallel hypersurfaces, Rend. Sem. Mat. Univ. Politechn. Torino 45 (1986), 303-316.

R. Deszcz, On pseudosymmetric spaces, Bull. Soc. Belg. Math. Ser. A 44 (1992), 1-34.

F. Dillen, Semiparallel hypersurfaces of a real space form, Israel J. Math. 75 (1991), 193-202.

H. Endo, Invariant submanifolds in contact metric manifolds, Proc. Estonian Acad. Sci. Phys. Math. 39 (1990), 1-8.

M. Kon, Invariant submanifolds of normal contact metric manifolds, Kodai Math. Sem. Rep., 27 (1973), 330-336.

M. Kon, Invariant submanifolds in Sasakian manifolds, Math. Ann. 219 (1976), 277-290.

O. Kowalski, An explicit classification of 3-dimensional Riemannian spaces satisfying (R(X, Y )cdot R = 0), Czechoslov. Math. J. 46 (1996), 427-474.

U. Lumiste, Semisymmetric submanifolds as second order envelope of a symmetric submanifolds., Tensor (N.S) 43(1) (1986), 83-87.

B.C. Montano, Di Terlizzi and Mukut Mani Tripathi, Invariant submanifolds of contact ((k,mu))-manifolds, Glassgow Math J., 50 (2008), 499-507.

C. Murathan, K. Arslan and E. Ezentas, Ricci generalized pseudo-symmetric immersions, Diff. Geom.Appl. (2005), 99-108.

C. Ozgur and C. Murathan, On invariant submanifolds of Lorenzian para-Sasakian manifolds, The Arabian J. Sci. Engg., 34(2A) (2009), 177-185.

P.A. Shirokov, Constant vector fields and tensor fields of second order in Riemannian spaces, Izv. Kazan Fiz. Mat. Obshchestva Ser. 25(2) (1925), 86114.

S. Sular and C. Ozgur, On some submanifolds of Kenmotsu manifolds, Chaos, Solitons and Fractals. 42 (2009), 1990-1995.

Z.I. Szabo, Structure theorems on Riemannian manifolds satisfying (R(X, Y )cdot R = 0), I Local version. J. Differ. Geom. 17 (1982), 531-582.

Z.I. Szabo, Structure theorems on Riemannian manifolds satisfying (R(X, Y )cdot R = 0), II Global version. Geom. Dedicata 19 (1985), 65-108.

S. Tanno, Ricci curvatures of contact Riemannian manifolds, Tohoku Mathematical Journal 40, 441 (1988).

M.M. Tripathi, T. Sasahara and J.S. Kim, On invariant submanifolds of contact metric manifolds, Tsukuba J. Math. 29(2) (2005), 495-510.

L. Verstraelen,, Comments on pseudosymmetry in the sense of Ryszard Deszcz, In Geometry and Topology of submanifolds of World Scientific Publishing, River Edge vol. VI (1994), 199-209.

K. Yano, Concircular geometry I. concircular transformations in:, Proceedings of the Imperial Academy Tokyo, 16 (1940), 195-200.

K. Yano and M. Kon, Structures on manifolds, World scientific publishing, (1984).

Ahmet Yildz and Cengizhan Murathan, Invariant submanifolds of Sasakian space forms, J. Geom. 95 (2009), 135-150.

Downloads

Published

2017-06-09
CITATION

How to Cite

Siddesha, M. S., & Bagewadi, C. S. (2017). On Some Classes of Invariant Submanifolds of \((k,\mu)\)-Contact Manifolds. Journal of Informatics and Mathematical Sciences, 9(1), 13–25. https://doi.org/10.26713/jims.v9i1.451

Issue

Section

Research Articles