Almost CoKähler manifolds satisfying Miao-Tam equation
- 58 Downloads
The aim of the present paper is to classify almost CoKähler manifolds satisfying Miao-Tam equation. We find the expression of the curvature tensor in an almost CoKähler manifold of dimension greater than 3 with \(\xi \) belonging to the \((k, \mu )\)-nullity distribution and \(k<0\). We prove that gradient of \(\lambda \) is pointwise collinear with \(\xi \). As a consequence, we obtain that the potential function \(\lambda \) is constant. Finally, we show that the solution of the Miao-Tam equation on almost CoKähler manifolds of dimension greater than 3 with \(\xi \) belonging to the \((k, \mu )\)-nullity distribution and \(k<0\) is either trivial or Einstein.
KeywordsHessian Laplacian Miao-Tam equation almost CoKähler manifolds
Mathematics Subject Classification53C15 53C25
The authors are thankful to the reviewer for his/her valuable suggestions for the better improvement of the paper. Also the author Debabrata Kar is supported by CSIR, India (File no: 09/028(1007)/2017-EMR-1).
- 3.Blair, D.E.: Contact manifold in Riemannian Geometry, Lecture Notes on Mathematics, Springer, Berlin (1976)Google Scholar
- 4.Blair, D.E.: Riemannian Geometry on contact and sympletic manifolds, Progress in Mathematics, Birkhäuser, Boston, 203 (2010)Google Scholar
- 7.Dacko, P., Olszak, Z.: On almost cosympletic \((k,\mu ,\nu )\) -space. Banach Center Publ. 69, 211–220 (2005)Google Scholar
- 8.Dey, D., Majhi, P.: Some critical metrics on 3-dimensional trans-Sasakian manifolds. Palestine J. Math. (2018) (Accepted) Google Scholar