[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.7468/jksmeb.2013.20.3.199

A NOTE ON SCALAR CURVATURE FUNCTIONS OF ALMOST-KÄHLER METRICS

Kim, Jongsu (Department of Mathematics, Sogang University)

Publication Information

The Pure and Applied Mathematics / v.20, no.3, 2013 , pp. 199-206 More about this Journal

Abstract

We present a 4-dimensional nil-manifold as the first example of a closed non-K $\ddot{a}$ hlerian symplectic manifold with the following property: a function is the scalar curvature of some almost K $\ddot{a}$ hler metric iff it is negative somewhere. This is motivated by the Kazdan-Warner's work on classifying smooth closed manifolds according to the possible scalar curvature functions.

Keywords

almost K $\ddot{a}$ hler metric; scalar curvature; symplectic manifold;

Citations & Related Records

Reference

1	N. Hitchin: On compact four-dimensional Einstein manifolds. J. Diff. Geom. 9 (1974), 435-442.
2	W. Thurston: Some simple examples of symplectic manifolds. Proc. Amer. Math. Soc. 55 (1976), 467-468.
3	A.L. Besse: Einstein manifolds. Ergebnisse der Mathematik, 3 Folge, Band 10, Springer-Verlag, 1987.
4	D.E. Blair: On the set of metrics associated to a symplectic or contact form, Bull. Institute of Math. Academia Sinica. 11 (1983), 297-308.
5	R.E. Gompf & A.I. Stipsicz: 4-Manifolds and Kirby Calculus. Graduate Studies in Math. v.20, American Math. Soc., 1999
6	M. Gromov: Pseudoholomorphic curves in symplectic manifolds. Invent. Math. 82 (1985), 307-347. DOI
7	M. Gromov & H.B. Lawson: Spin and scalar curvature in the presence of a fundamental group. I. Ann. of Math. 111 (1980), no. 2, 209-230. DOI
8	J.L. Kazdan & F.W. Warner: Existence and conformal deformation of metrics with prescribed gaussian and scalar curvatures. Ann. of Math. 101 (1975), no. 2, 317-331. DOI ScienceOn
9	D. McDuff & D. Salamon: Introduction to Symplectic Topology. Oxford University Press, New York 1998.
10	J. Kim & C. Sung: Deformation of almost Kahler metrics with constant scalar curvature on compact Kahler manifolds. Ann. Global. Anal. and Geom. 22 (2002), 49-73. DOI
11	R. Schoen: Conformal deformation of a Riemannian metric to constant scalar curvature. J. Diff. Geom. 20 (1984), 479-495.
12	T. Aubin: Some nonlinear problems in Riemannian geometry. Springer Monographs in Mathematics, Springer-Verlag, 1998.
13	E. Abbena: An example of an almost Kahler manifold which is not Kahlerian. Boll. Un. Mat. Ital. A 3 (1984), no. 6, 383-392.