Certain results on N(k)-contact metric manifolds

Uday Chand De


In the present paper we study contact metric manifolds whose characteristic vector field $\xi$ belonging to the $k$-nullity distribution. First we consider concircularly pseudosymmetric $N(k)$-contact metric manifolds of dimension $(2n+1)$. Beside these, we consider Ricci solitons and gradient Ricci solitons on three dimensional $N(k)$-contact metric manifolds. As a consequence we obtain several results. Finally, an example is given.


Three dimensional N(k)-contact metric manifolds; concircularly pseudosymmetric; Ricci soliton; Gradient Ricci soliton; Einstein manifolds; Three dimensional N(k)-contact metric manifolds; concircularly pseudosymmetric; Ricci soliton

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C. L. Bejan and M. Crasmareanu,Ricci solitons in manifolds with quasi--constant curvature, Publ. Math. Debrecen, 78(1) (2011), 235--243.

D. E. Blair, Contact manifolds in Riemannian geometry, Lecture note in Math., 509, Springer--Verlag, Berlin--New York, 1976.

D. E. Blair, T. Koufogiorgos and R. Sharma, A classification of $3$--dimensional contact metric manifolds with $Qphi =phi Q$, Kodai Math. J., 13(1990), 391--401.

D. E. Blair, J. S. Kim and M. M. Tripathi,On the concircular curvature tensor of a contact metric manifold, J. Korean Math. Soc., 42 5(2005), 883--992.

E. Boeckx, O. Kowalski and L. Vanhecke,Riemannian manifolds of conullity two, Singapore World Sci. Publishing, 1996.

E. Boeckx, A full classification of contact metric $(k,mu)$--spaces, Illinois J. Math., 44(2010), 212--219.

E. Cartan, Sur une classe remarqable d' espaces de Riemannian, Bull. Soc. Math. France., 54(1962), 214--264.

U. C. De, C. Murathan and K. Arsalan, On the Weyl projective curvature tensor of an $N(k)$--contact metric manifold, Mathematica Panonoica, 21(1) (2010), 129--142.

U. C. De and A. Sarkar, On the quasi--conformal curvature tensor of a $(k,mu)$--contact metric manifold, Math. Reports, 14(64), 2 (2012), 115--129.

U. C. De, A. Yildiz and S. Ghosh, On a class of $N(k)$--contact metric manifolds, Math. Reports., 16(66)(2014), No. 2.

Avik De and J. B. Jun, On $N(k)$-contact metric manifolds satisfying certain curvature conditions, Kyungpook Math. J., 51, 4 (2011), 457--468.

J. B. Jun and U. K. Kim, On $3$--dimensional almost contact metric manifolds, Kyungpook Math. J., 34(2) (1994), 293--301.

O. Kowalski, An explicit classification of $3$-- dimensional Riemannian spaces satisfying $R(X,Y).R=0$, Czechoslovak Math. J., 46, 121 (1996), 427--474.

C. Calin and M. Crasmareanu, From the Eisenhart problem to Ricci solitons in f-Kenmotsu manifolds, Bull. Malays. Math. Soc., 33(3)(2010), 361--368.

J. T. Cho, Notes on contact Ricci solitons, Proc. Edinb. Math. Soc., 54(2011), 47--53.

T. Chave and G. Valent, Quasi--Einstein metrics and their renoirmalizability properties, Helv. Phys. Acta., 69(1996), 344--347.

T. Chave and G. Valent, On a class of compact and non--compact quasi--Einstein metrics and their renoirmalizability properties, Nuclear Phys. B., 478(1996), 758--778.

B. Y. Chen, Some results on concircular vector fields and their applications to Ricci solitons, Bull. Korean Math. Soc., 52(2015), 1535--1547.

B. Y. Chen and S. Deshmukh, Ricci solitons and concurrent vector fields, Balkan J. Geom. Appl., 20(2015), 14--25.

B. Y. Chen and S. Deshmukh,Classification of Ricci solitons on Euclidean hypersurfaces,Internat. J. Math., 25(2014), no.~11, 1450104, 22 pp.

Chen, B. Y. and Deshmukh, S., Geometry of compact shrinking Ricci solitons ., Balkan J. Geom. Appl. 19 (2014), no. 1, 13--21.

B. Chow and D. Knoff, The Ricci flow: An introduction, Mathematical surveys and Monographs 110, American Math. Soc., 2004.

U. C. De and S. Biswas, A note on $xi $--conformally flat contact manifolds, Bull. Malays. Math. Soc., 29(2006), 51--57.

U. C. De and A. K. Mondal, Three dimensional Quasi--Sasakian manifolds and Ricci solitons, SUT J. Math., 48(1)(2012), 71--81.

U. C. De and P. Majhi, On a type of contact metric manifolds, Lobacheviskii J. Math., 34(2013), 89--98.

A. Derdzinski, Compact Ricci solitons, Preprint.

A. Derdzinski, A Myers--type theorem and compact Ricci solitons, Proc. Amer. Math. Soc., 134(12)(2006), 3645--3648.

S. Deshmukh, Jacobi--type vector fields on Ricci solitons, Bull. Math. Soc. Sci. Math. Roumanie, 55(103), 1 (2012), 41--50.

S. Deshmukh, H. Alodan and H. Al-Sodais, A Note on Ricci Soliton, Balkan J. Geom. Appl., 16(1)(2011), 48--55.

D. Friedan, Non linera models in $2+epsilon$ dimensions, Ann. Phys.,163 (1985), 318--419.

R. S. Hamilton, The Ricci flow on surfaces, Mathematics and general relativity

(Santa Cruz, CA, 1986), 237--262. Contemp. Math., 71, American Math. Soc., 1988.

A. Ghosh, R. Sharma and J. T. Cho, Contact metric manifolds with $eta$-parallel torsion tensor, Ann. Global Anal. Geom.,34(3), 287--299.

T. Ivey, Ricci solitons on compact 3--manifolds, Diff. Geom. Appl., 3(1993), 301--307.

T. Ivey, New examples of complete Ricci solitons, Proc. Amer. Math. Soc., 122(1) (1994), 241--245.

P. Majhi and U. C. De, Classifications of $N(k)$-contact metric manifolds satisfying certain curvature conditions,Acta Math. Univ. Commenianae, LXXXIV(1)(2015), 167--178.

C. Ozgur, Contact metric manifolds with cyclic--parallel Ricci tensor, Diff. Geom. Dynamical Systems, 4 (2002), 21--25.

C. Ozgur and S. Sular, On $N(k)$-contact metric manifolds satisfying certain conditions, SUT J. Math., 44(1)(2008), 89--99.

G. Perelman, The entropy formula for the Ricci flow and its geometric applications, Preprint, http://arXiv.org/abs/Math.DG/0211159.

Z. I.Szabo,Structure theorems on Riemannian spaces satisfying $R(X,Y).R=0$, the local version, J. Diff. Geom., 17(1982), 531--582.

R. Sharma, Certain results on $K$-contact and $(k,mu)$-contact manifolds, J. of Geometry, 89(2008), 138--147.

L. Verstraelen,Comments on pseudosymmetry in the sense of Ryszard Deszcz, In: Geometry and Topology of submanifolds, VI. River Edge, NJ: World Sci. Publishing, 1994, 199--209.

A. Taleshian and A. A. Hosseinzabeh, Investigation of Some Conditions on $N(k)$-Quasi Einstein Manifolds,Bull. Malays. Math. Soc., 34(3) (2011), 455--464.

K. Yano and S. Bochner, Curvature and Betti numbers, Annals of mathematics studies, 32, Princeton university press, 1953.

K. Yano, Concircular geometry I. concircular transformations, Proc. Imp. Acad. Tokyo, 16(1940), 195--200.

Tanno, S., Ricci curvature of contact Riemannian manifolds, Tohoku Math. J., 40(1988), 441--448.

W. Wylie, Complete shrinking Ricci solitons have finite fundamental group, Proc. Amer. Math. Soc., 136(5) (2008), 1803--1806.

DOI: http://dx.doi.org/10.5556/j.tkjm.49.2018.2506

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