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Three-dimensional Riemannian Manifolds Associated with Locally Conformal Riemannian Product Manifolds

Year 2025, Volume: 18 Issue: 1, 1 - 13, 24.04.2025

İva Dokuzova

Abstract

A 3-dimensional Riemannian manifold equipped with a tensor structure of type (1,1), whose fourth power is the identity, is considered. This structure acts as an isometry with respect to the metric. A Riemannian almost product manifold associated with such a manifold is also studied. It turns out, that the almost product manifold belongs to the class of locally conformal Riemannian product manifolds of the Naveira classification. Conditions for the additional structures of the manifolds to be parallel with respect to the Levi-Civita connection of the metric were found. Classes of almost Einstein manifolds and Einstein manifolds are determined and some of their curvature properties are obtained. As examples of these manifolds, a hypersurface is considered.

Keywords

Riemannian manifold locally product structure Einstein manifold Ricci curvature hypersurface

References

  • Alegre, P., Carriazo, A.: Curves as slant submanifolds of an almost product Riemannian manifold, Turkish J. Math. 48 (4), 701–712 (2024). https://doi.org/10.55730/1300-0098.3535
  • Atçeken, M., Keleş, S.: On the product Riemannian manifolds, Differ. Geom. Dyn. Syst. 5 (1), 1–8 (2003).
  • Aydin, M.E., López, R.: Rotational surfaces in $R^4$ with new approaches and examples, Int. Electron. J. Geom. 17 (1), 97–105 (2024). https://doi.org/10.36890/iejg.1430210
  • Dokuzova, I.: Four-dimensional Riemannian product manifolds with circulant structures, Stud. Univ. Babeç-Bolyai Math. 68 (2), 439–448 (2023). https://doi.org/10.24193/subbmath.2023.2.17
  • Erkan, E., Takano, K., Gülbahar, M.: Locally product-like statisical manifolds and their hypersurfaces, Int. Electron. J. Geom. 16 (2), 435–450 (2023). https://doi.org/10.36890/iejg.1307467
  • Gribacheva, D.:A natural connection on a basic class of Riemannian product manifolds, Int. J. Geom. Methods Mod. Phys. 9 (7), no. 1250057, 14pp. (2012). https://doi.org/10.1142/S0219887812500570
  • Gribacheva, D.: Curvature properties of two Naveira classes of Riemannian product manifolds, Plovdiv. Univ. Paisiı Khilendarski Nauchn. Trud. Mat. 39 (3), 31–42 (2012). Preprint arXiv:1204.5838
  • Gribacheva, D., Mekerov, D.: Natural connections on conformal Riemannian P-manifolds, C. R. Acad. Bulgare Sci. 65 (5), 581–590 (2012).
  • Gribacheva, D., Mekerov, D.: Conformal Riemannian P-manifolds with connections whose curvature tensors are Riemannan P-tensors, J. Geom. 105 (2), 273–286 (2014). https://doi.org/10.1007/s00022-013-0206-y
  • Manev, M., Tavkova, V.: Hyperspheres in Euclidean and Minkowski 4-spaces as almost paracontact almost paracomplex Riemannian manifolds, Novi Sad J. Math. 51 (2), 175–185 (2021). https://doi.org/10.30755/NSJOM.12136
  • Naveira, A.M.: A classification of Riemannian almost product manifolds, Rend. Mat. (7) 3 (3), 577–592 (1983).
  • Pitiş, Gh.: On some submanifolds of a locally product manifold, Kodai Math. J. 9 (4), 327–333 (1986).
  • Pušic, N.: On some connections on locally product Riemannian manifolds - Part I, Novi Sad J. Math. 41 (2), 29–40 (2011).
  • Razpopov, D., Dzhelepov, G.: Curvature properties of Riemannian manifolds with circulant structures, Adv. Math. Sci. J. 9 (1), 37–47 (2020). https://doi.org/10.37418/amsj.9.1.4
  • Razpopov, D., Dzhelepov, G.: A work done by an isotropic vector force field along an isotropic curve, Balkan J. Geom. Appl. 26 (1), 69–80 (2021).
  • Rashevsky, P.K.: Riemannian geometry and tensor analysis, Nauka Eds., Moscow (1967).
  • Salimov, A.A., Iscan, M., Akbulut, K.: Some remarks concerning hyperholomorphic B-manifolds, Chinese Ann. Math. Ser. B 29 (6), 631–640 (2008). https://doi.org/10.1007/s11401-007-0441-3.
  • Salimov A.A., Gezer, A., Iscan, M.: On para-Këhler-Norden structures on the tangent bundles, Ann. Polon. Math. 103 (3), 247–261 (2012). https://doi.org/10.4064/ap103-3-3.
  • Staikova, M., Gribachev, K.: Canonical connections and their conformal invariants on Riemannian almost product-manifolds, Serdica 18 (3-4), 150-161 (1992).
  • Yano, K.: Differential geometry on complex and almost complex spaces, International Series of Monographs in Pure and Applied Mathematics 49, The Macmillan Company, New York (1965).

Year 2025, Volume: 18 Issue: 1, 1 - 13, 24.04.2025

İva Dokuzova

Abstract

References

  • Alegre, P., Carriazo, A.: Curves as slant submanifolds of an almost product Riemannian manifold, Turkish J. Math. 48 (4), 701–712 (2024). https://doi.org/10.55730/1300-0098.3535
  • Atçeken, M., Keleş, S.: On the product Riemannian manifolds, Differ. Geom. Dyn. Syst. 5 (1), 1–8 (2003).
  • Aydin, M.E., López, R.: Rotational surfaces in $R^4$ with new approaches and examples, Int. Electron. J. Geom. 17 (1), 97–105 (2024). https://doi.org/10.36890/iejg.1430210
  • Dokuzova, I.: Four-dimensional Riemannian product manifolds with circulant structures, Stud. Univ. Babeç-Bolyai Math. 68 (2), 439–448 (2023). https://doi.org/10.24193/subbmath.2023.2.17
  • Erkan, E., Takano, K., Gülbahar, M.: Locally product-like statisical manifolds and their hypersurfaces, Int. Electron. J. Geom. 16 (2), 435–450 (2023). https://doi.org/10.36890/iejg.1307467
  • Gribacheva, D.:A natural connection on a basic class of Riemannian product manifolds, Int. J. Geom. Methods Mod. Phys. 9 (7), no. 1250057, 14pp. (2012). https://doi.org/10.1142/S0219887812500570
  • Gribacheva, D.: Curvature properties of two Naveira classes of Riemannian product manifolds, Plovdiv. Univ. Paisiı Khilendarski Nauchn. Trud. Mat. 39 (3), 31–42 (2012). Preprint arXiv:1204.5838
  • Gribacheva, D., Mekerov, D.: Natural connections on conformal Riemannian P-manifolds, C. R. Acad. Bulgare Sci. 65 (5), 581–590 (2012).
  • Gribacheva, D., Mekerov, D.: Conformal Riemannian P-manifolds with connections whose curvature tensors are Riemannan P-tensors, J. Geom. 105 (2), 273–286 (2014). https://doi.org/10.1007/s00022-013-0206-y
  • Manev, M., Tavkova, V.: Hyperspheres in Euclidean and Minkowski 4-spaces as almost paracontact almost paracomplex Riemannian manifolds, Novi Sad J. Math. 51 (2), 175–185 (2021). https://doi.org/10.30755/NSJOM.12136
  • Naveira, A.M.: A classification of Riemannian almost product manifolds, Rend. Mat. (7) 3 (3), 577–592 (1983).
  • Pitiş, Gh.: On some submanifolds of a locally product manifold, Kodai Math. J. 9 (4), 327–333 (1986).
  • Pušic, N.: On some connections on locally product Riemannian manifolds - Part I, Novi Sad J. Math. 41 (2), 29–40 (2011).
  • Razpopov, D., Dzhelepov, G.: Curvature properties of Riemannian manifolds with circulant structures, Adv. Math. Sci. J. 9 (1), 37–47 (2020). https://doi.org/10.37418/amsj.9.1.4
  • Razpopov, D., Dzhelepov, G.: A work done by an isotropic vector force field along an isotropic curve, Balkan J. Geom. Appl. 26 (1), 69–80 (2021).
  • Rashevsky, P.K.: Riemannian geometry and tensor analysis, Nauka Eds., Moscow (1967).
  • Salimov, A.A., Iscan, M., Akbulut, K.: Some remarks concerning hyperholomorphic B-manifolds, Chinese Ann. Math. Ser. B 29 (6), 631–640 (2008). https://doi.org/10.1007/s11401-007-0441-3.
  • Salimov A.A., Gezer, A., Iscan, M.: On para-Këhler-Norden structures on the tangent bundles, Ann. Polon. Math. 103 (3), 247–261 (2012). https://doi.org/10.4064/ap103-3-3.
  • Staikova, M., Gribachev, K.: Canonical connections and their conformal invariants on Riemannian almost product-manifolds, Serdica 18 (3-4), 150-161 (1992).
  • Yano, K.: Differential geometry on complex and almost complex spaces, International Series of Monographs in Pure and Applied Mathematics 49, The Macmillan Company, New York (1965).
There are 20 citations in total.

Details

Primary Language English
Subjects Algebraic and Differential Geometry
Journal Section Research Article
Authors

İva Dokuzova 0000-0002-9099-6969

Early Pub Date April 20, 2025
Publication Date April 24, 2025
Submission Date September 29, 2024
Acceptance Date November 28, 2024
Published in Issue Year 2025 Volume: 18 Issue: 1

Cite

APA Dokuzova, İ. (2025). Three-dimensional Riemannian Manifolds Associated with Locally Conformal Riemannian Product Manifolds. International Electronic Journal of Geometry, 18(1), 1-13.
AMA Dokuzova İ. Three-dimensional Riemannian Manifolds Associated with Locally Conformal Riemannian Product Manifolds. Int. Electron. J. Geom. April 2025;18(1):1-13.
Chicago Dokuzova, İva. “Three-Dimensional Riemannian Manifolds Associated With Locally Conformal Riemannian Product Manifolds”. International Electronic Journal of Geometry 18, no. 1 (April 2025): 1-13.
EndNote Dokuzova İ (April 1, 2025) Three-dimensional Riemannian Manifolds Associated with Locally Conformal Riemannian Product Manifolds. International Electronic Journal of Geometry 18 1 1–13.
IEEE İ. Dokuzova, “Three-dimensional Riemannian Manifolds Associated with Locally Conformal Riemannian Product Manifolds”, Int. Electron. J. Geom., vol. 18, no. 1, pp. 1–13, 2025.
ISNAD Dokuzova, İva. “Three-Dimensional Riemannian Manifolds Associated With Locally Conformal Riemannian Product Manifolds”. International Electronic Journal of Geometry 18/1 (April 2025), 1-13.
JAMA Dokuzova İ. Three-dimensional Riemannian Manifolds Associated with Locally Conformal Riemannian Product Manifolds. Int. Electron. J. Geom. 2025;18:1–13.
MLA Dokuzova, İva. “Three-Dimensional Riemannian Manifolds Associated With Locally Conformal Riemannian Product Manifolds”. International Electronic Journal of Geometry, vol. 18, no. 1, 2025, pp. 1-13.
Vancouver Dokuzova İ. Three-dimensional Riemannian Manifolds Associated with Locally Conformal Riemannian Product Manifolds. Int. Electron. J. Geom. 2025;18(1):1-13.
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