Show Summary Details
More options …

Advances in Pure and Applied Mathematics

Editor-in-Chief: Trimeche, Khalifa

Editorial Board: Aldroubi, Akram / Anker, Jean-Philippe / Aouadi, Saloua / Bahouri, Hajer / Baklouti, Ali / Bakry, Dominique / Baraket, Sami / Ben Abdelghani, Leila / Begehr, Heinrich / Beznea, Lucian / Bezzarga, Mounir / Bonami, Aline / Demailly, Jean-Pierre / Fleckinger, Jacqueline / Gallardo, Leonard / Ismail, Mourad / Jarboui, Noomen / Jouini, Elyes / Karoui, Abderrazek / Kamoun, Lotfi / Kobayashi, Toshiyuki / Maday, Yvon / Marzougui, Habib / Mili, Maher / Mustapha, Sami / Ovsienko, Valentin / Peigné, Marc / Pouzet, Maurice / Radulescu, Vicentiu / Schwartz, Lionel / Sifi, Mohamed / Zaag, Hatem / Zarati, Said

CiteScore 2017: 1.29

SCImago Journal Rank (SJR) 2017: 0.177
Source Normalized Impact per Paper (SNIP) 2017: 0.409

Mathematical Citation Quotient (MCQ) 2017: 0.21

Online
ISSN
1869-6090
See all formats and pricing
More options …
Volume 9, Issue 3

Some curvature properties of paracontact metric manifolds

Krishanu Mandal
• Corresponding author
• Department of Pure Mathematics, University of Calcutta, 35, Ballygunge Circular Road, Kolkata 700019, West Bengal, India
• Email
• Other articles by this author:
/ Uday Chand De
• Department of Pure Mathematics, University of Calcutta, 35, Ballygunge Circular Road, Kolkata 700019, West Bengal, India
• Email
• Other articles by this author:
Published Online: 2017-10-24 | DOI: https://doi.org/10.1515/apam-2017-0064

Abstract

The purpose of this paper is to study Ricci semisymmetric paracontact metric manifolds satisfying ${\nabla }_{\xi }h=0$ and such that the sectional curvature of the plane section containing ξ equals a non-zero constant c. Also, we study paracontact metric manifolds satisfying the curvature condition $Q\cdot R=0$, where Q and R are the Ricci operator and the Riemannian curvature tensor, respectively, and second order symmetric parallel tensors in paracontact metric manifolds under the same conditions. Several consequences of these results are discussed.

MSC 2010: 53B30; 53C15; 53C25; 53D10; 53D15

References

• [1]

G. Calvaruso, Homogeneous paracontact metric three-manifolds, Illinois J. Math. 55 (2011), no. 2, 697–718. Google Scholar

• [2]

B. Cappelletti Montano, Bi-paracontact structures and Legendre foliations, Kodai Math. J. 33 (2010), no. 3, 473–512.

• [3]

B. Cappelletti-Montano, A. Carriazo and V. Martín-Molina, Sasaki–Einstein and paraSasaki–Einstein metrics from $\left(\kappa ,\mu \right)$-structures, J. Geom. Phys. 73 (2013), 20–36.

• [4]

B. Cappelletti Montano and L. Di Terlizzi, Geometric structures associated to a contact metric $\left(\kappa ,\mu \right)$-space, Pacific J. Math. 246 (2010), no. 2, 257–292.

• [5]

B. Cappelletti Montano, I. Küpeli Erken and C. Murathan, Nullity conditions in paracontact geometry, Differential Geom. Appl. 30 (2012), no. 6, 665–693.

• [6]

U. C. De, Second order parallel tensors on P-Sasakian manifolds, Publ. Math. Debrecen 49 (1996), no. 1–2, 33–37. Google Scholar

• [7]

U. C. De, S. Deshmukh and K. Mandal, On three-dimensional $N\left(k\right)$-paracontact metric manifolds and Ricci solitons, Bull. Iranian Math. Soc., to appear. Google Scholar

• [8]

U. C. De and K. Mandal, Certain results on generalized $\left(k,\mu \right)$-contact metric manifolds, J. Geom. 108 (2017), no. 2, 611–621.

• [9]

L. P. Eisenhart, Symmetric tensors of the second order whose first covariant derivatives are zero, Trans. Amer. Math. Soc. 25 (1923), no. 2, 297–306.

• [10]

S. Kaneyuki and F. L. Williams, Almost paracontact and parahodge structures on manifolds, Nagoya Math. J. 99 (1985), 173–187.

• [11]

H. Levy, Symmetric tensors of the second order whose covariant derivatives vanish, Ann. of Math. (2) 27 (1925), no. 2, 91–98.

• [12]

V. Martín-Molina, Local classification and examples of an important class of paracontact metric manifolds, Filomat 29 (2015), no. 3, 507–515.

• [13]

V. A. Mirzoyan, Structure theorems for Riemannian Ric-semisymmetric spaces, Izv. Vyssh. Uchebn. Zaved. Mat. (1992), no. 6, 80–89. Google Scholar

• [14]

R. Sharma, Second order parallel tensors on contact manifolds, Algebras Groups Geom. 7 (1990), no. 2, 145–152. Google Scholar

• [15]

P. A. Shirokov, Constant vector fields and tensor fields of second order in Riemannian spaces, Izv. Kazan Fiz. Mat. Obshchestva Ser 25 (1925), 86–114. Google Scholar

• [16]

Z. I. Szabó, Structure theorems on Riemannian spaces satisfying $R\left(X,Y\right)\cdot R=0$. I. The local version, J. Differential Geom. 17 (1982), no. 4, 531–582. Google Scholar

• [17]

P. Verheyen and L. Verstraelen, A new intrinsic characterization of hypercylinders in Euclidean spaces, Kyungpook Math. J. 25 (1985), no. 1, 1–4. Google Scholar

• [18]

Y. Wang and X. Liu, Second order parallel tensors on almost Kenmotsu manifolds satisfying the nullity distributions, Filomat 28 (2014), no. 4, 839–847.

• [19]

S. Zamkovoy, Canonical connections on paracontact manifolds, Ann. Global Anal. Geom. 36 (2009), no. 1, 37–60.

Revised: 2017-09-16

Accepted: 2017-10-02

Published Online: 2017-10-24

Published in Print: 2018-07-01

Citation Information: Advances in Pure and Applied Mathematics, Volume 9, Issue 3, Pages 159–165, ISSN (Online) 1869-6090, ISSN (Print) 1867-1152,

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.