CRITICAL POINT METRICS OF THE TOTAL SCALAR CURVATURE

Chang, Jeong-Wook; Hwang, Seung-Su; Yun, Gab-Jin

doi:10.4134/BKMS.2012.49.3.655

HOME > Journal Browse > About Journal > Journal Vol & Issue > Full Record

CRITICAL POINT METRICS OF THE TOTAL SCALAR CURVATURE

Journal title : Bulletin of the Korean Mathematical Society
Volume 49, Issue 3, 2012, pp.655-667
Publisher : The Korean Mathematical Society
DOI : 10.4134/BKMS.2012.49.3.655

Title & Authors

CRITICAL POINT METRICS OF THE TOTAL SCALAR CURVATURE
Chang, Jeong-Wook; Hwang, Seung-Su; Yun, Gab-Jin;

Abstract

In this paper, we deal with a critical point metric of the total scalar curvature on a compact manifold ${$M$}$ . We prove that if the critical point metric has parallel Ricci tensor, then the manifold is isometric to a standard sphere. Moreover, we show that if an ${$n$}$ -dimensional Riemannian manifold is a warped product, or has harmonic curvature with non-parallel Ricci tensor, then it cannot be a critical point metric.

Keywords

the total scalar curvature;critical point metric;Einstein;

Language

English

Cited by

1time(s) in KSCD

THREE DIMENSIONAL CRITICAL POINT OF THE TOTAL SCALAR CURVATURE,;

대한수학회보, 2013. vol.50. 3, pp.867-871

3time(s) in CrossRef

A note on critical point metrics of the total scalar curvature functional, Journal of Mathematical Analysis and Applications, 2015, 424, 2, 1544

THREE DIMENSIONAL CRITICAL POINT OF THE TOTAL SCALAR CURVATURE, Bulletin of the Korean Mathematical Society, 2013, 50, 3, 867

Critical metrics of the total scalar curvature functional on 4-manifolds, Mathematische Nachrichten, 2015, 288, 16, 1814

References

A. L. Besse, Einstein Manifolds, Springer-Verlag, Berlin, Heidelberg, 1987.

R. L. Bishop and B. O'Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 1-49.

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

A. Derdzinski, On compact Riemannian manifolds with harmonic curvature, Math. Ann. 259 (1982), no. 2, 145-152.

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

A. Gray, Einstein-like manifolds which are not Einstein, Geom. Dedicata 7 (1978), no. 3, 259-280.

S. Hwang, 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.

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

S. Hwang and J. W. Chang, The Critical point equation on a four dimensional warped product manifold, Bull. Korean Math. Soc. 43 (2006), no. 4, 679-692.

10.

O. Kobayashi, A differential equation arising from scalar curvature function, J. Math. Soc. Japan 34 (1982), no. 4, 665-675.

11.

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

12.

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

13.

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

14.

B. O'Neill, Semi-Riemannian Geometry, Academic Press, New York, 1983.