THE CRITICAL POINT EQUATION ON A FOUR DIMENSIONAL WARPED PRODUCT MANIFOLD

Title & Authors
THE CRITICAL POINT EQUATION ON A FOUR DIMENSIONAL WARPED PRODUCT MANIFOLD
Hwang, Seung-Su; Chang, Jeong-Wook;

Abstract
On a compact oriented n-dimensional manifold $\small{(M^n,\;g)}$, it has been conjectured that a metric g satisfying the critical point equation (2) should be Einstein. In this paper, we prove that if a manifold $\small{(M^4,\;g)}$ is a 4-dimensional oriented compact warped product, then g can not be a solution of CPE with a non-zero solution function f.
Keywords
critical point equation;warped product;Einstein metric;
Language
English
Cited by
1.
CRITICAL POINT METRICS OF THE TOTAL SCALAR CURVATURE,;;;

대한수학회보, 2012. vol.49. 3, pp.655-667
1.
CRITICAL POINT METRICS OF THE TOTAL SCALAR CURVATURE, Bulletin of the Korean Mathematical Society, 2012, 49, 3, 655
References
1.
A. L. Besse, Einstein Manifolds, Springer-Verlag, New York, 1987

2.
J. P. Bourguignon, Une stratification de l'espace des structures riemanniennes, Compositio Math. 30 (1975), 1-41

3.
A. E. Fischer and J. E. Marsden, Manifolds of Riemannian metrics with prescribed scalar curvature, Bull. Amer. Math. Soc. 80 (1974), 479-484

4.
S. Hwang, Critical points of the total scalar curvature functionals on the space of metrics of constant scalar curvature, Manuscripta Math. 103 (2000), no. 2, 135-142

5.
S. Hwang , The critical point equation on a three dimensional compact manifold, Proc. Amer. Math. Soc. 131 (2003), no. 10, 3221-3230

6.
S. Hwang and J. W. Chang, Critical points and warped product metrics, Bull. Korean Math. Soc. 41 (2004), no. 1, 117-123

7.
J. L. Kazdan and F. W. Warner, Curvature functions for open 2-manifolds, Ann. Math. (2) 99 (1974), 203-219

8.
J. Lafontaine, Remarques sur les varietes conformement plates, Math. Ann. 259 (1982), no. 3, 313-319

9.
J. Lafontaine, Sur la geometrie dune generalisation de lequation differentielle d'Obata, J. Math. Pures Appl. 62 (1983), no. 1, 63-72

10.
M. Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan 14 (1962), no. 3, 333-340

11.
B. ONeill, Semi-Riemannian Geometry, Academic Press, 1983