Critical point equation on almost Kenmotsu manifolds
Keywords:
Almost Kenmotsu manifold, nullity distribution, critical point equation, Einstein manifoldAbstract
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 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.
References
Barros, Abdênago; Ribeiro, Ernani, Jr. Critical point equation on four-dimensional compact manifolds. Math. Nachr. 287 (2014), no. 14-15, 1618--1623. doi: 10.1002/mana.201300149
Besse, Arthur L. Einstein manifolds. Reprint of the 1987 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2008. xii+516 pp. ISBN: 978-3-540-74120-6 doi: 10.1007/978-3-540-74311-8
Blair, David E. Contact manifolds in Riemannian geometry. Lecture Notes in Mathematics, Vol. 509. Springer-Verlag, Berlin-New York, 1976. {rm vi}+146 pp. MR0467588
Blair, David E. Riemannian geometry of contact and symplectic manifolds. Second edition. Progress in Mathematics, 203. Birkhouser Boston, Ltd., Boston, MA, 2010. xvi+343 pp. ISBN: 978-0-8176-4958-6 doi: 10.1007/978-0-8176-4959-3
De, U. C.; Mandal, Krishanu. On phi-Ricci recurrent almost Kenmotsu manifolds with nullity distributions. Int. Electron. J. Geom. 9 (2016), no. 2, 70--79. MR3571433
De, U. C.; Mandal, Krishanu. On a type of almost Kenmotsu manifolds with nullity distributions. Arab J. Math. Sci. 23 (2017), no. 2, 109--123. doi: 10.1016/j.ajmsc.2016.04.001
De, Uday Chand; Mandal, Krishanu. On locally phi-conformally symmetric almost Kenmotsu manifolds with nullity distributions. Commun. Korean Math. Soc. 32 (2017), no. 2, 401--416. doi: 10.4134/CKMS.c160073
Dileo, Giulia; Pastore, Anna Maria. Almost Kenmotsu manifolds and local symmetry. Bull. Belg. Math. Soc. Simon Stevin 14 (2007), no. 2, 343--354. MR2341570
Dileo, Giulia; Pastore, Anna Maria. Almost Kenmotsu manifolds and nullity distributions. J. Geom. 93 (2009), no. 1-2, 46--61. doi: 10.1007/s00022-009-1974-2
Ghosh, Amalendu; Patra, Dhriti Sundar. The critical point equation and contact geometry. J. Geom. 108 (2017), no. 1, 185--194. doi: 10.1007/s00022-016-0333-3
Hwang, Seungsu. Critical points of the total scalar curvature functional on the space of metrics of constant scalar curvature. Manuscripta Math. 103 (2000), no. 2, 135--142. doi: 10.1007/PL00005857
Vanhecke, Lieven; Janssens, Dirk. Almost contact structures and curvature tensors. Kodai Math. J. 4 (1981), no. 1, 1--27. 10.2996/kmj/1138036310
Kenmotsu, Katsuei. A class of almost contact Riemannian manifolds. Tohoku Math. J. (2) 24 (1972), 93--103. doi: 10.2748/tmj/1178241594
Neto, Benedito Leandro. A note on critical point metrics of the total scalar curvature functional. J. Math. Anal. Appl. 424 (2015), no. 2, 1544--1548. doi: 10.1016/j.jmaa.2014.11.040
Pastore, Anna Maria; Saltarelli, Vincenzo. Almost Kenmotsu manifolds with conformal Reeb foliation. Bull. Belg. Math. Soc. Simon Stevin 18 (2011), no. 4, 655--666. MR2907610
Wang, Yaning; Liu, Ximin. Second order parallel tensors on almost Kenmotsu manifolds satisfying the nullity distributions. Filomat 28 (2014), no. 4, 839--847. doi: 10.2298/FIL1404839W
Wang, Yaning; Liu, Ximin. Riemannian semisymmetric almost Kenmotsu manifolds and nullity distributions. Ann. Polon. Math. 112 (2014), no. 1, 37--46. doi: 10.4064/ap112-1-3
Wang, Yaning; Liu, Ximin. On a type of almost Kenmotsu manifolds with harmonic curvature tensors. Bull. Belg. Math. Soc. Simon Stevin 22 (2015), no. 1, 15--24. MR3325717
Wang, Yaning; Liu, Ximin. On almost Kenmotsu manifolds satisfying some nullity distributions. Proc. Nat. Acad. Sci. India Sect. A 86 (2016), no. 3, 347--353. doi: 10.1007/s40010-016-0274-0
