• P-ISSN 0974-6846 E-ISSN 0974-5645

Indian Journal of Science and Technology


Indian Journal of Science and Technology

Year: 2023, Volume: 16, Issue: Special Issue 1, Pages: 174-178

Original Article

On Generalized Quasi-conformally 2-recurrent Riemannian Manifolds

Received Date:23 January 2023, Accepted Date:09 June 2023, Published Date:13 September 2023


Objectives: The object of the present paper is to study a type of Riemannian manifold called generalized quasi-conformally 2-recurrent Riemannian manifold, it is denoted by G { e C (2Kn )} . Methods: Differentiable manifold was defined on the basis of topology, differential calculus and real analysis. Riemannian manifold is a part of differentiable manifold in which we study by tensor notation and index free notation. Using such notations, several types of recurrent manifold and their standard results are used to characterize generalized quasi-conformally 2-recurrent Riemannian manifold and also several results of tensor calculus such as contraction, covariant differentiation, divergence of tensor, Bianchi’s identities, Ricci identity, Fundamental theorem of Riemannian geometry and many more are used in this paper to finding the interesting results. Finding: We study G { e C (2Kn )} manifold with Einstein manifold and prove that an Einstein G { e C (2Kn )} is a manifold of constant curvature. We also established a necessary and sufficient condition for an EinsteinG { e C (2Kn )} manifold to be a generalized 2-recurrent Riemannian manifold G { e C (2Kn )} . In the last, we used vanishes Ricci tensor and shows that G { e C (2Kn )} manifold reduces toG { e C (2Kn )} . Novelty: Recurrent manifold and generalized 2- recurrent Riemannian manifold has already been studied in published literature. In this paper we study generalized 2-recurrent Riemannian manifold with quasi-conformal curvature tensor and find interesting results.

Keywords: Recurrent manifold; Ricci tensor; generalized 2-recurrent manifold; quasi-conformal curvature tensor; Einstein manifold


  1. Walker AG. On Ruse's Spaces of Recurrent Curvature. Proceedings of the London Mathematical Society. 1950;s2-52(1):36–64. Available from: https://doi.org/10.1112/plms/s2-52.1.36
  2. Cartan E. Sur une classe remarquable d'espaces de Riemann. Bulletin de la Société Mathématique de France. 1926;54:214–264. Available from: https://doi.org/10.24033/bsmf.1105
  3. Courbure LA. Courbure, Nombres de Betti, et Espaces Symétriques. Proceedings of the International Congress of Mathematicians. 1950;2:216–223. Available from: https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1950.2/ICM1950.2.ocr.pdf
  4. Roy AK. On generalized 2-recurrent tensor in Riemannian space. Bulletins de l'Académie Royale de Belgique. 1972;58:220–228. Available from: https://doi.org/10.3406/barb.1972.60446
  5. Singh H, Sinha R. On Special Curvature Tensor in a Generalized 2-recurrent Smooth Riemannian Manifold. Journal of the Tensor Society. 2010;4(01):49–55. Available from: https://doi.org/10.56424/jts.v4i01.10424
  6. Shaikh AA, Patra A. On a generalized class of recurrent manifolds. Archivum Mathematicum (BRNO). 2010;46:71–78. Available from: https://www.emis.de/journals/AM/10-1/am1798.pdf
  7. Yano K, Sawaki S. Riemannian manifolds admitting a conformal transformation group. Journal of Differential Geometry. 1968;2(2):161–184. Available from: https://doi.org/10.4310/jdg/1214428253
  8. L P Eisenhart. Riemannian Geometry. (pp. 1-316) Princeton University Press. 1949.
  9. Bishop RL, Goldberg SI. On Conformally Flat Spaces with Commuting Curvature and Ricci Transformations. Canadian Journal of Mathematics. 1972;24(5):799–804. Available from: https://doi.org/10.4153/cjm-1972-077-6
  10. Jaishwal JP, Ojha RH. On Generalized ϕ-Recurrent LP-Sasakian Manifolds. Kyungpook Mathematical Journal. 2009;49:779–788. Available from: https://koreascience.kr/article/JAKO200923437149410.pdf
  11. UCD, Sarkar A. On Three-Dimensional Locally Φ-Recurrent Quasi-Sasakian Manifolds. Demonstratio Mathematica. 2008;41(3):677–684. Available from: https://doi.org/10.1515/dema-2008-0319
  12. Ingalahalli G, Bagewadi CS. A Study on ϕ-recurrence τ-curvature tensor in (k, µ)-contact metric manifolds. Communications in Mathematics. 2018;26(2):1–10. Available from: https://doi.org/10.2478/cm-2018-0009


© 2023 Kumar. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Published By Indian Society for Education and Environment (iSee)


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