LOCALLY CONFORMAL KÄHLER MANIFOLDS AND CONFORMAL SCALAR CURVATURE

Title & Authors
LOCALLY CONFORMAL KÄHLER MANIFOLDS AND CONFORMAL SCALAR CURVATURE
Kim, Jae-Man;

Abstract
We show that on a compact locally conformal K$\small{\ddot{a}}$hler manifold $\small{M^{2n}}$ (dim $\small{M^{2n}\;=\;2n\;{\geq}\;4}$), $\small{M^{2n}}$ is K$\small{\ddot{a}}$hler if and only if its conformal scalar curvature k is not smaller than the scalar curvature s of $\small{M^{2n}}$ everywhere. As a consequence, if a compact locally conformal K$\small{\ddot{a}}$hler manifold $\small{M^{2n}}$ is both conformally flat and scalar flat, then $\small{M^{2n}}$ is K$\small{\ddot{a}}$hler. In contrast with the compact case, we show that there exists a locally conformal K$\small{\ddot{a}}$hler manifold with k equal to s, which is not K$\small{\ddot{a}}$hler.
Keywords
compact locally conformal K$\small{\ddot{a}}$hler manifold;conformal scalar curvature;K$\small{\ddot{a}}$hler;conformally flat and scalar flat;a locally conformal K$\small{\ddot{a}}$hler manifold with k equal to s;
Language
English
Cited by
References
1.
V. Apostolov and P. Gauduchon, The Riemannian Goldberg-Sachs theorem, Internat. J. Math. 8 (1997), no. 4, 421-439.

2.
A. L. Besse, Einstein Manifolds, Springer-Verlag, Berlin, 1987.

3.
S. Dragomir and L. Ornea, Locally Conformal Kahler Geometry, Progress in Mathematics, 155. Birkhauser Boston, Inc., Boston, MA, 1998.

4.
P. Gauduchon and S. Ivanov, Einstein-Hermitian surfaces and Hermitian Einstein-Weyl structures in dimension 4, Math. Z. 226 (1997), no. 2, 317-326.

5.
P. Gauduchon, La 1-forme de torsion d'une variete hermitienne compacte, Math. Ann. 267 (1984), no. 4, 495-518.

6.
Z. Hu, H. Li, and U. Simon, Schouten curvature functions on locally conformally flat Riemannian manifolds, J. Geom. 88 (2008), no. 1-2, 75-100.

7.
J. Kim, Rigidity theorems for Einstein-Thorpe metrics, Geom. Dedicata 80 (2000), no. 1-3, 281-287.

8.
J. Kim, On Einstein Hermitian manifolds, Monatsh. Math. 152 (2007), no. 3, 251-254.

9.
I. Vaisman, Some curvature properties of complex surfaces, Ann. Mat. Pura Appl. (4) 132 (1982), 1-18.

10.
I. Vaisman, Some curvature properties of locally conformal Kahler manifolds, Trans. Amer. Math. Soc. 259 (1980), no. 2, 439-447.

11.
I. Vaisman, A theorem on compact locally conformal Kahler manifolds, Proc. Amer. Math. Soc. 75 (1979), no. 2, 279-283.