Abstract
We provide some properties of Riemann solitons with torse-forming potential vector fields, pointing out their relation to Ricci solitons. We also study those Riemann soliton submanifolds isometrically immersed into a Riemannian manifold endowed with a torse-forming vector field, having as potential vector field its tangential component. We consider the minimal and the totally geodesic cases, too, as well as when the ambient manifold is of constant sectional curvature. In particular, we prove that a totally geodesic submanifold isometrically immersed into a Riemannian manifold endowed with a concircular vector field is a Riemann soliton if and only if it is of constant curvature. Furthermore, we show that, if the potential vector field of a minimal hypersurface Riemann soliton isometrically immersed into a Riemannian manifold of constant curvature and endowed with a concircular vector field is of constant length, then it is a metallic shaped hypersurface.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availability
Data sharing is not applicable to this article as no new data were created or analyzed in this study.
References
Biswas, G.G., Chen, X., De, U.C.: Riemann solitons on almost co-Kähler manifolds. Filomat 36, 1403–1413 (2022). https://doi.org/10.2298/FIL2204403B
Blaga, A.M.: Remarks on almost Riemann solitons with gradient or torse-forming vector field. Bull. Malaysian Math. Sci. Soc. 44, 3215–3227 (2021). https://doi.org/10.1007/s40840-021-01108-9
Blaga, A.M., Laţcu, D.R.: Remarks on Riemann and Ricci solitons in \((\alpha,\beta )\)-contact metric manifolds. J. Geom. Symm. Phys. 58, 1–12 (2020). https://doi.org/10.7546/jgsp-58-2020-1-12
Blaga, A.M., Özgür, C.: Almost \(\eta \)-Ricci and almost \(\eta \)-Yamabe solitons with torse-forming potential vector field. Quaestiones Mathematicae 45, 143–163 (2022). https://doi.org/10.2989/16073606.2020.1850538
Blaga, A.M., Özgür, C. Remarks on submanifolds as almost \(\eta \)-Ricci-Bourguignon solitons, Facta Universitatis. Series: Mathematics and Informatics, 37, 397–407 (2020). https://doi.org/10.22190/FUMI220318027B
Chen, B.-Y.: A survey on Ricci solitons on Riemannian submanifolds, Recent advances in the geometry of submanifolds – dedicated to the memory of Franki Dillen (1963–2013), 27–39, Contemp. Math. 674, Amer. Math. Soc., Providence, RI, 2016
Chen, B.-Y., Deshmukh, S.: Ricci solitons and concurrent vector fields. Balkan J. Geom. Appl. 20, 14–25 (2015)
Chen, B.-Y., Yano, K.: Hypersurfaces of a conformally flat space. Tensor NS 26, 318–322 (1972)
De, K., De, U.C.: A note on almost Riemann solitons and gradient almost Riemann solitons. Afr. Mat. 33(74), 10 (2022). https://doi.org/10.1007/s13370-022-01010-y
De, K., De, U.C.: Riemann solitons on para-Sasakian geometry. Carpathian Math. Publ. 14, 395–405 (2022)
Devaraja, M.N., Kumara, H.A., Venkatesha, V.: Riemann soliton within the framework of contact geometry. Quaestiones Mathematicae 44, 637–651 (2021). https://doi.org/10.2989/16073606.2020.1732495
Gowda, P.D., Naik, D.M., Ravindranatha, A.M., Venkatesha, V.: Riemann solitons on \((\kappa, \mu )\)-almost cosymplectic manifolds. Commun. Korean Math. Soc. 38, 881–892 (2023). https://doi.org/10.4134/CKMS.c220243
Manev, M.: Almost Riemann solitons with vertical potential on conformal cosymplectic contact complex Riemannian manifolds. Symmetry 15, 104 (2023). https://doi.org/10.3390/sym15010104
Özgür, C., Özgür, N.Y.: Classification of metallic shaped hypersurfaces in real space forms. Turk. J. Math. 39, 784–794 (2015). https://doi.org/10.3906/mat-1408-17
Hamilton, R.S.: Three-manifolds with positive Ricci curvature. J. Differential Geom. 17, 255–306 (1982). https://doi.org/10.4310/jdg/1214436922
Hirică, I.E., Udrişte, C.: Ricci and Riemann solitons. Balkan J. Geom. Appl. 21, 35–44 (2016)
Pigola, S., Rigoli, M., Rimoldi, M., Setti, A.G.: Ricci almost solitons. Ann. Sc. Norm. Super. Pisa Cl. Sci. 5(10), 757–799 (2011)
Rachunek, L., Mikes, J.: On tensor fields semiconjugated with torse-forming vector fields. Acta Univ. Palacki. Olomuc. Fac. Rerum Nat. Math. 44, 151–160 (2005)
Tokura, W., Barboza, M., Batista, E., Menezes, I.: Rigidity results for Riemann and Schouten solitons. Mediterr. J. Math. 20, 112 (2023). https://doi.org/10.1007/s00009-023-02319-z
Venkatesha, V., Kumara, H.A., Naik, D.M.: Riemann solitons and almost Riemann solitons on almost Kenmotsu manifolds. Int. J. Geom. Methods Modern Phys. 17(2050105), 22 (2020). https://doi.org/10.1142/S0219887820501054
Yano, K.: On the torse-forming directions in Riemannian spaces. Proc. Imp. Acad. Tokyo 20, 340–345 (1944). https://doi.org/10.3792/pia/1195572958
Funding
No financial support was received for this manuscript.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors have no competing interests to declare that are relevant to the content of this article.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Communicated by Pablo Mira.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Blaga, A.M., Özgür, C. On Submanifolds as Riemann Solitons. Bull. Malays. Math. Sci. Soc. 47, 63 (2024). https://doi.org/10.1007/s40840-024-01661-z
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40840-024-01661-z