We study the critical point equation (CPE) conjecture on almost Kenmotsu manifolds. First, we prove that if a three-dimensional (k, μ)'-almost Kenmotsu manifold satisfies the CPE, then the manifold is either locally isometric to the product space ℍ2(−4) × ℝ or the manifold is a Kenmotsu manifold. Further, we prove that if the metric of an almost Kenmotsu manifold with conformal Reeb foliation satisfies the CPE conjecture, then the manifold is Einstein.
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References
A. Barros and E. Ribeiro (Jr.), “Critical point equation on four-dimensional compact manifolds,” Math. Nachr., 287, 1618–1623 (2014).
A. Besse, Einstein Manifolds, Springer, New York (2008).
D. E. Blair, “Contact manifold in Riemannian geometry,” Lect. Notes Math., 509 (1976).
D. E. Blair, “Riemannian geometry on contact and symplectic manifolds,” Progr. Math., 203 (2010).
U. C. De and K. Mandal, “On 𝜙-Ricci recurrent almost Kenmotsu manifolds with nullity distributions,” Int. Electron. J. Geom., 9, 70–79 (2016).
U. C. De and K. Mandal, “On a type of almost Kenmotsu manifolds with nullity distributions,” Arab J. Math. Sci., 23, 109–123 (2017).
U. C. De and K. Mandal, “On locally 𝜙-conformally symmetric almost Kenmotsu manifolds with nullity distributions,” Comm. Korean Math. Soc., 32, 401–416 (2017).
G. Dileo and A. M. Pastore, “Almost Kenmotsu manifolds and local symmetry,” Bull. Belg. Math. Soc. Simon Stevin, 14, 343–354 (2007).
G. Dileo and A. M. Pastore, “Almost Kenmotsu manifolds and nullity distributions,” J. Geom., 93, 46–61 (2009).
A. Ghosh and D. S. Patra, “The critical point equation and contact geometry,” J. Geom., 108, 185–194 (2017).
S. Hwang, “Critical points of the total scalar curvature functionals on the space of metrics of constant scalar curvature,” Manuscripta Math., 103, 135–142 (2000).
D. Janssens and L. Vanhecke, “Almost contact structures and curvature tensors,” Kodai Math. J., 4, 1–27 (1981).
K. Kenmotsu, “A class of almost contact Riemannian manifolds,” Tohoku Math. J., 24, 93–103 (1972).
B. L. Neto, “A note on critical point metrics of the total scalar curvature functionals,” J. Math. Anal. Appl., 424, 1544–1548 (2015).
A. M. Pastore and V. Saltarelli, “Almost Kenmotsu manifolds with conformal Reeb foliation,” Bull. Belg. Math. Soc. Simon Stevin, 18, 655–666 (2011).
Y. Wang and X. Liu, “Second order parallel tensors on almost Kenmotsu manifolds satisfying the nullity distributions,” Filomat, 28, 839–847 (2014).
Y. Wang and X. Liu, “Riemannian semisymmetric almost Kenmotsu manifolds and nullity distributions,” Ann. Polon. Math., 112, 37–46 (2014).
Y.Wang and X. Liu, “On a type of almost Kenmotsu manifolds with harmonic curvature tensors,” Bull. Belg. Math. Soc. Simon Stevin, 22, 15–24 (2015).
Y. Wang and X. Liu, “On almost Kenmotsu manifolds satisfying some nullity distributions,” Proc. Nat. Acad. Sci. India. Sect. A., 86, 347–353 (2016).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 1, pp. 61–68, January, 2020
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De, U.C., Mandal, K. Critical Point Equation on Almost Kenmotsu Manifolds. Ukr Math J 72, 69–77 (2020). https://doi.org/10.1007/s11253-020-01770-5
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DOI: https://doi.org/10.1007/s11253-020-01770-5