Critical point equation on almost Kenmotsu manifolds

  • U. C. De Univ. Calcutta, West Bengal, India
  • K.  Mandal Univ. Calcutta, West Bengal, India
Keywords: Almost Kenmotsu manifold, nullity distribution, critical point equation, Einstein manifold

Abstract

We study the critical point equation $(CPE)$ conjecture on almost Kenmotsu manifolds.
First, we prove that if a three-dimensional $(k,\mu)'$-almost Kenmotsu manifold satisfies the $CPE,$ then the manifold is either locally isometric to the product space $\mathbb H^2(-4)\times\mathbb R$ 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

Published
15.01.2020
How to Cite
De U. C., and Mandal K. “Critical Point Equation on Almost Kenmotsu Manifolds”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 1, Jan. 2020, pp. 61-68, http://umj.imath.kiev.ua/index.php/umj/article/view/2330.
Section
Research articles