Weak Quasi-Contact Metric Manifolds and New Characteristics of K-Contact and Sasakian Manifolds
Abstract
:1. Introduction
- (1)
- —normal a.c.m. manifolds, characterized by the equality .
- (2)
- —Sasakian manifolds, characterized by the equality; see ([3] Theorem 6.3),
- (3)
- —contact metric manifolds, characterized by the equality , where .
- (4)
- —nearly-Sasakian manifolds, characterized by the equality; see [4]:
- (5)
- —quasi-contact metric (q.c.m.) manifolds, characterized by the equality
- (1)
- belongs to if and only if belongs to —Hermitian manifolds, defined by .
- (2)
- belongs to if and only if belongs to —Kähler manifolds, defined by , where is the Levi–Civita connection for .
- (3)
- belongs to if and only if belongs to —almost-Kähler manifolds, defined by , where .
- (4)
- belongs to if and only if belongs to —nearly-Kähler manifolds, defined by .
- (5)
- belongs to if and only if belongs to —quasi-Kähler manifolds, defined by .
2. Preliminaries
3. Main Results
4. Conclusions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
a.c.m | almost-contact metric |
q.c.m | quasi-contact metric |
References
- Chinea, D.; Gonzalez, C. A classification of almost contact metric manifolds. Ann. Mat. Pura Appl. 1990, 156, 15–36. [Google Scholar] [CrossRef]
- Gray, A.; Hervella, L.M. The sixteen classes of almost Hermitian manifolds and their linear imvarients. Ann. Mat. Pura Appl. 1980, 123, 35–58. [Google Scholar] [CrossRef]
- Blair, D.E. Riemannian Geometry of Contact and Symplectic Manifolds, 2nd ed.; Springer: New York, NY, USA, 2010. [Google Scholar]
- Blair, D.E.; Showers, D.K.; Komatu, Y. Nearly Sasakian structures. Kodai Math. Sem. Rep. 1976, 27, 175–180. [Google Scholar] [CrossRef]
- Bae; Park, J.; H, J.; Sekigawa, K. Quasi contact metric manifolds with Killing characteristic vector fields. Bull. Korean Math. Soc. 2020, 57, 1299–1306. [Google Scholar]
- Chai, Y.D.; Kim, J.H.; Park, J.H.; Sekigawa, K.; Shin, W.M. Notes on quasi contact metric manifolds. An. Ştiinţ Univ. Al. I. Cuza Iaşi Mat. (N.S.) 2016, 1, 349–359. [Google Scholar]
- Kim, J.H.; Park, J.H.; Sekigawa, K. A generalization of contact metric manifolds. Balkan J. Geom. Appl. 2014, 19, 94–105. [Google Scholar]
- Patra, D.S.; Rovenski, V. On the rigidity of the Sasakian structure and characterization of cosymplectic manifolds. Differ. Geom. Its Appl. 2023, 90, 102043. [Google Scholar] [CrossRef]
- Rovenski, V.; Wolak, R. New metric structures on g-foliations. Indag. Math. 2022, 33, 518–532. [Google Scholar] [CrossRef]
- Rovenski, V. Generalized Ricci solitons and Einstein metrics on weak K-contact manifolds. Commun. Anal. Mech. 2023, 15, 177–188. [Google Scholar] [CrossRef]
- Rovenski, V. Weak nearly Sasakian and weak nearly cosymplectic manifolds. Mathematics 2023, 11, 4377. [Google Scholar] [CrossRef]
- Rovenski, V. On the splitting of weak nearly cosymplectic manifolds. Differ. Geom. Its Appl. 2024, 94, 102142. [Google Scholar] [CrossRef]
- Rovenski, V. Characterization of Sasakian manifolds. Asian-Eur. J. Math. 2024, 17, 2450030. [Google Scholar] [CrossRef]
- Rovenski, V. Weak almost contact structures: A survey. arXiv 2024, arXiv:2408.13827. [Google Scholar]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2024 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Rovenski, V. Weak Quasi-Contact Metric Manifolds and New Characteristics of K-Contact and Sasakian Manifolds. Mathematics 2024, 12, 3230. https://doi.org/10.3390/math12203230
Rovenski V. Weak Quasi-Contact Metric Manifolds and New Characteristics of K-Contact and Sasakian Manifolds. Mathematics. 2024; 12(20):3230. https://doi.org/10.3390/math12203230
Chicago/Turabian StyleRovenski, Vladimir. 2024. "Weak Quasi-Contact Metric Manifolds and New Characteristics of K-Contact and Sasakian Manifolds" Mathematics 12, no. 20: 3230. https://doi.org/10.3390/math12203230
APA StyleRovenski, V. (2024). Weak Quasi-Contact Metric Manifolds and New Characteristics of K-Contact and Sasakian Manifolds. Mathematics, 12(20), 3230. https://doi.org/10.3390/math12203230