SavedAdd your journal

Journals / Turkish Journal of Mathematics / 2015 / Cilt: 39 - Sayı: 3

On some classes of $3$-dimensional generalized $ (\kappa ,\mu )$-contact metric manifolds

Ahmet YILDIZ
Uday Chand DE
Azime ÇETİNKAYA
Pages
356–368
Publication Date
2015-08-13
DOI
—
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Finally, we construct an example of a generalized $(\kappa,\mu)$-contact metric manifold to verify Theorem $1$ of our paper.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:The object of the present paper is to obtain a necessary and sufficient condition for a $3$-dimensional generalized $(\kappa ,\mu )$-contact metric manifold to be locally $\phi $-symmetric in the sense of Takahashi and the condition is verified by an example.
📊
Key Findings & Data:Next we characterize a $3$-dimensional generalized $(\kappa ,\mu )$-contact metric manifold satisfying certain curvature conditions on the concircular curvature tensor.
💡
Conclusion & Impact:Finally, we construct an example of a generalized $(\kappa,\mu)$-contact metric manifold to verify Theorem $1$ of our paper.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Generalized $(\kappa · \mu )$-contact metric manifolds · concircular curvature tensor

Abstract

The object of the present paper is to obtain a necessary and sufficient condition for a $3$-dimensional generalized $(\kappa ,\mu )$-contact metric manifold to be locally $\phi $-symmetric in the sense of Takahashi and the condition is verified by an example. Next we characterize a $3$-dimensional generalized $(\kappa ,\mu )$-contact metric manifold satisfying certain curvature conditions on the concircular curvature tensor. Finally, we construct an example of a generalized $(\kappa,\mu)$-contact metric manifold to verify Theorem $1$ of our paper.

Özet

The object of the present paper is to obtain a necessary and sufficient condition for a $3$-dimensional generalized $(\kappa ,\mu )$-contact metric manifold to be locally $\phi $-symmetric in the sense of Takahashi and the condition is verified by an example. Next we characterize a $3$-dimensional generalized $(\kappa ,\mu )$-contact metric manifold satisfying certain curvature conditions on the concircular curvature tensor. Finally, we construct an example of a generalized $(\kappa,\mu)$-contact metric manifold to verify Theorem $1$ of our paper.

🔖

Cite This Article

YILDIZ, A., DE, U. C., ÇETİNKAYA, A. (2015). On some classes of $3$-dimensional generalized $ (\kappa ,\mu )$-contact metric manifolds. Turkish Journal of Mathematics, Cilt: 39 - Sayı: 3, 356–368. https://ajindex.com/articles/723307

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (15)

  1. Blair DE. Contact manifolds in Riemannian geometry. New York, NY, USA: Springer-Verlag, 1976. Blair DE. Two remarks on contact metric structures. Tohoku Math J 1977; 29: 319–324. Blair DE. Riemannian geometry of contact and symplectic manifolds. Boston, MA, USA: Birkhauser, 2002.
  2. Blair DE, Chen H. A classification of 3-dimensional contact metric manifolds with Qϕ = ϕQ. Bull Math Acad Sinica 1992; 20: 379–383.
  3. Blair DE, Kim JS, Tripathi MM. On the concircular curvature tensor of a contact metric manifold. Korean J Math Soc 2005; 42: 883–892.
  4. Blair DE, Koufogiorgos T, Papantoniou BJ. Contact metric manifolds satisfying a nullity condition. Israel J Math 1995; 91: 189–214.
  5. Cabrerizo JL, Fernandez LM, Fernandez M. The structure of a class of K -contact manifolds. Acta Math Hung 1999; 82: 331–340.
  6. Carriazo A, Martin-Molina V. Generalized (κ, µ) -space forms and Da-homothetic deformations. Balkan J Geo Appl 2011; 16: 59–69.
  7. Ghosh S, De UC. On a class of generalized (κ, µ) -contact metric manifolds. P Jangjeon Math Soc 2010; 13: 337–347.
  8. Gouli-Andreou F, Xenos PJ. A class of contact metric 3-manifolds with ξ∈ N(κ, µ) and κ, µ functions. Algebras Groups and Geometries 2000; 17: 401–407.
  9. Jin JE, Park JH, Sekigawa K. Notes on some classes of 3-dimensional contact metric manifolds. Balkan J Geo Appl 2012; 17: 54–65.
  10. Koufogiorgos T, Tsichlias C. On the existence of a new class contact metric manifolds. Can Math Bull 2000; 43: 440–447.
  11. Koufogiorgos T, Tsichlias C. Generalized (κ, µ) -contact metric manifolds with|| gradκ ||= constant. J Geo 2003; 78: 83–91. Papantoniou BJ. Contact Riemannian manifolds satisfying R(ξ, X).R = 0 and ξ∈ (κ, µ) -nullity distribution. Yokohama Math J 1993; 40: 149–161. Perrone D. Weakly ϕ -symmetric contact metric spaces, Balkan J Geo Appl 2002; 7: 67–77.
  12. Sarkar A, De UC, Sen M. Some results on generalized (κ, µ) -contact metric manifolds. Acta Universitatis Apulensis 2012; 32: 49–59.
  13. Shaikh AA, Arslan K, Murathan C, Baishya KK. On 3-dimensional generalized (κ, µ) -contact metric manifolds. Balkan J Geo Appl 2007; 2: 122–134. Takahashi T. Sasakian ϕ -symmetric spaces. Tohoku Math J 1977; 29: 91–113.
  14. Tripathi MM, Kim JS. On the concircular curvature tensor of a (κ, µ) -manifold. Balkan J Geo Appl 2004; 2: 104–114. Yano K. Concircular geometry I. Concircular transformations. P Imp Acad 1940; 16: 195–200.
  15. Yano K, Bochner S. Curvature and Betti Numbers. Princeton, NJ, USA: Princeton University Press, 1953. Zhen G. On conformal symmetric K -cont