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F-TRACELESS COMPONENT OF THE CONFORMAL CURVATURE TENSOR ON KÄHLER MANIFOLD

  • Funabashi, Shoichi (DEPARTMENT OF MATHEMATICS NIPOPON INSTITUTE OF TECHNOLOGY) ;
  • Kim, Hang-Sook (DEPARTMENT OF COMPUTATIONAL MATHEMATICS SCHOOL OF COMPUTER AIDED SCIENCE AND INSTITUTE OF MATHEMATICAL SCIENCES COLLEGE OF NATURAL SCIENCE, INJE UNIVERSITY) ;
  • Kim, Young-Mi (DEPARTMENT OF COMPUTATIONAL MATHEMATICS SCHOOL OF COMPUTER AIDED SCIENCE AND INSTITUTE OF MATHEMATICAL SCIENCES COLLEGE OF NATURAL SCIENCE, INJE UNIVERSITY) ;
  • Pak, Jin-Suk (DEPARTMENT OF MATHEMATICS EDUCATION KYUNGPOOK NATIONAL UNIVERSITY)
  • Published : 2007.11.30

Abstract

We investigate F-traceless component of the conformal curvature tensor defined by (3.6) in $K\ddot{a}hler$ manifolds of dimension ${\geq}4$, and show that the F-traceless component is invariant under concircular change. In particular, we determine $K\ddot{a}hler$ manifolds with parallel F-traceless component and improve some theorems, provided in the previous paper([2]), which are concerned with the traceless component of the conformal curvature tensor and the spectrum of the Laplacian acting on $p(0{\leq}p{\leq}2)$-forms on the manifold by using the F-traceless component.

Keywords

$K\ddot{a}hler$ manifold;conformal curvature tensor;traceless decomposition;F-traceless decomposition;constant holomorphic sectional curvature;spectrum

References

  1. M. Berger, P. Gauduchon, and E. Mazet, Le Spectre d'une Variete riemannienne, Lecture Notes in Mathematics 194, Springer-Verlag, 1971
  2. S. Funabashi, H. S. Kim, Y.-M. Kim, and J. S. Pak, Traceless component of the confor- mal curvature tensor in Kahler manifold, Czech. Math. J. 56 (2006), 857-874 https://doi.org/10.1007/s10587-006-0061-1
  3. H. Kitahara, K. Matsuo, and J. S. Pak, Appendium; A conformal curvature tensor field on hermitian manifolds, Bull. Korean Math. Soc. 27 (1990), 27-30
  4. D. Krupka, The trace decomposition problem, Beitrage Algebra Geom. 36 (1995), no. 2, 303-315
  5. L. Lakoma and M. Jukl, The decomposition of tensor spaces with almost complex struc- ture, Rend. Circ. Mat. Palermo (2) Suppl. No. 72 (2004), 145-150
  6. L. Mikes, On general trace decomposition problem, Proc. Conf., Aug. 28-Sept. 1, Brno, Czech Rep. (1995), 45-50
  7. V. K. Patodi, Curvature and the fundamental solution of the heat operator, J. Indian Math. Soc. 34 (1970), no. 3-4, 269-285
  8. S. Tachibana, Riemannian geometry, Asakura Shoten, Tokyo, 1967
  9. S. Tanno, Eigenvalues of the Laplacian of Riemannian manifolds, Tohoku Math. J. (2) 25 (1973), 391-403 https://doi.org/10.2748/tmj/1178241341
  10. Gr. Tsagas, On the spectrum of the Laplace operator for the exterior 2-forms, Tensor(N.S.) 33 (1979), no. 1, 94-96
  11. S. Yamaguchi and G. Chuman, Eigenvalues of the Laplacian of Sasakian manifolds, TRU Math. 15 (1979), no. 2, 31-41
  12. K. Yano, Differential geometry on complex and almost complex spaces, International Series of Monographs in Pure and Applied Mathematics, Vol. 49, A Pergamon Press Book, The Macmillan Co., New York, 1965
  13. H. Kitahara, K. Matsuo, and J. S. Pak, A conformal curvature tensor field on hermitian manifolds, J. Korean Math. Soc. 27 (1990), 7-17
  14. J. S. Pak, K.-H. Cho, and J.-H. Kwon, Conformal curvature tensor field and spectrum of the Laplacian in Kaehlerian manifolds, Bull. Korean Math. Soc. 32 (1995), no. 2, 309-319