Abstract
Let \((M^{2n+1},\phi ,\xi ,\eta ,g)\) be a non-Kenmotsu almost Kenmotsu \((k,\mu )'\)-manifold. If the metric g represents a Yamabe soliton, then either \(M^{2n+1}\) is locally isometric to the product space \(\mathbb {H}^{n+1}(-4)\times \mathbb {R}^n\) or \(\eta \) is a strict infinitesimal contact transformation. The later case can not occur if a Yamabe soliton is replaced by a gradient Yamabe soliton. Some corollaries of this theorem are given and an example illustrating this theorem is constructed.
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References
Blair, D.E.: Riemannian geometry of contact and symplectic manifolds, Progress in Mathematics, vol. 203. Birkhäuser, Basel (2010)
Chen, X.: Almost quasi-Yamabe solitons on almost cosymplectic manifolds. Int. J. Geom. Methods Mod. Phys. 17, 2050070 (2020) (p. 16)
Dai, X., Zhao, Y., De, U.C.: \(\star \)-Ricci soliton on \((\kappa,\mu )^{\prime }\)-almost Kenmotsu manifolds. Open Math. 17, 874–882 (2019)
Dey, C., De, U.C.: A note on quasi-Yamabe solitons on contact metric manifolds. J. Geom. 111, 11 (2020)
Dileo, G., Pastore, A.M.: Almost Kenmotsu manifolds and nullity distributions. J. Geom. 93, 46–61 (2009)
Ghosh, A.: Yamabe soliton and quasi Yamabe soliton on Kenmotsu manifold. Math. Slovaca 70, 151–160 (2020)
Ghosh, A., Patra, D.S.: \(*\)-Ricci soliton within the frame-work of Sasakian and \((\kappa ,\mu )\)-contact manifold. Int. J. Geom. Methods Modern Phys. 15, 1850120 (2018) (p. 21)
Hamilton, R.S.: The Ricci flow on surfaces, mathematics and general relativity, contemporary mathematics, vol. 71, pp. 237–262. American Mathematical Society (1988)
Janssens, D., Vanhecke, L.: Almost contact structures and curvature tensors. Kodai Math. J. 4, 1–27 (1981)
Hui, S.K., Mandal, Y.C.: Yamabe solitons on Kenmotsu manifolds. Commun. Korean Math. Soc. 34, 321–331 (2019)
Kenmotsu, K.: A class of almost contact Riemannian manifolds. Tôhoku Math. J. 24, 93–103 (1972)
Majhi, P., De, U.C., Suh, Y.J.: \(*\)-Ricci solitons on Sasakian \(3\)-manifolds. Publ. Math. Debr. 93, 241–252 (2018)
Pastore, A.M., Saltarelli, V.: Generalized nullity conditions on almost Kenmotsu manifolds. Int. Electr. J. Geom. 4, 168–183 (2011)
Sharma, R.: A \(3\)-dimensional Sasakian metric as a Yamabe soliton. Int. J. Geom. Methods Mod. Phys. 9, 1220003 (2012) (p. 5)
Suh, Y.J., De, U.C.: Yamabe solitons and Ricci solitons on almost co-Kähler manifolds. Can. Math. Bull. 62, 653–661 (2019)
Tanno, S.: Note on infinitesimal transformations over contact manifolds. Tohoku Math. J. 14, 416–430 (1962)
Venkatesha, V., Naik, D.M.: Yamabe solitons on \(3\)-dimensional contact metric manifolds with \(Q\phi =\phi Q\). Int. J. Geom. Methods Mod. Phys. 16, 1950039 (2019) (p. 9)
Venkatesha, V., Naik, D.M., Kumara, H.A.: \(*\)-Ricci solitons and gradient almost \(*\)-Ricci solitons on Kenmotsu manifolds. Math. Slovaca 69, 1447–1458 (2019)
Wang, Y.: Yamabe solitons on three-dimensional Kenmotsu manifolds. Bull. Belg. Math. Soc. Simon Stevin 23, 345–355 (2016)
Wang, Y.: Contact \(3\)-manifolds and \(*\)-Ricci soliton. Kodai Math. J. 43, 256–267 (2020)
Wang, Y., Liu, X.: On almost Kenmotsu manifolds satisfying some nullity distributions. Proc. Natl. Acad. Sci. India Sect. A Phys. Sci. 86, 347–353 (2016)
Wang, Y., Wang, W.: Some results on \((k,\mu )^{\prime }\)-almost Kenmotsu manifolds. Quaest. Math. 41, 469–481 (2018)
Yano, K.: Integral formulas in Riemannian geometry. Marcel Dekker, New York (1970)
Acknowledgements
This work was supported by the Key Scientific Research Program in Universities of Henan Province (No. 20A110023) and the Fostering Foundation of National Foundation in Henan Normal University (No. 2019PL22). The author would like to thank the referees for their careful reading and many suggestions that improves the paper.
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Wang, Y. Almost Kenmotsu \((k,\mu )'\)-manifolds with Yamabe solitons. RACSAM 115, 14 (2021). https://doi.org/10.1007/s13398-020-00951-y
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DOI: https://doi.org/10.1007/s13398-020-00951-y