• Published : 2006.02.01


We investigate a curvature-like tensor defined by (3.1) in Sasakian manifold of $dimension{\geq}$ 5, and show that this tensor satisfies some properties. Especially, we determine compact Sasakian manifolds with vanishing this tensor and improve some theorems concerning contact conformal curvature tensor and spectrum of Laplacian acting on $p(0{\leq}P{\leq}2)-forms$ on the manifold by using this tensor component.


Sasakian manifold;curvature-like tensor field;D-homothetic;spectrum


  1. M. Berger, P. Gauduchonm, and E. Mazet, Le Spectre d'une Variete riemanni- enne, 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 conformal curvature tensor in Kahler manifold, preprint
  3. M. Matsumoto and G. chuman, On the C-Bochner curvature tensor, TRU Math. 5 (1969), 21-30
  4. K. Ogiue, On almost contact manifolds admitting axiom of planes or axioms of free mobility, Kodai Math. Sem. Rep 16 (1964), 223-232
  5. J. S. Pak, J. C. Jeong, and W. -T. Kim, The contact conformal curvature tensor field and the spectrum of the Laplacian, J. Korean Math. Soc. 28 (1991), no. 2, 267-274
  6. V. K. Patodi, Curvature and the fundamental solution of the heat operator, J. Indian Math. Soc. 34 (1970), no. 3-4, 269-285
  7. S. Sasaki, Almost contact manifolds, Lecture Notes I, II, Tohoku Univ, 1965
  8. S. Tanno, Sasakian manifolds with constant $\phi$-holomorphic sectional curvature, Tohoku Math. J. (2) 21 (1969), 501-507
  9. S. Yamaguchi and G. Chuman, Eigenvalues of the Laplacian of Sasakian man- ifolds, TRU Math. 15 (1979), no. 2, 31-41
  10. S. Tanno, Eigenvalues of the Laplacian of Riemannian manifolds, Tohoku Math. J. 25 (1973), 391-403
  11. J. C. Jeong, J. D. Lee, G. H. Oh, and J. S. Pak, On the contact conformal curvature tensor, Bull. Korean Math. Soc. 27 (1990), no. 2, 133-142
  12. W. M. Boothby and H. C. Wang, On contact manifolds, Ann. of Math (2) 68 (1958), 721-734
  13. Gr. Tsagas, On the spectrum of the Laplace operator for the exterior 2-forms, Tensor (N. S.) 33 (1979), no. 1, 94-96