# A CURVATURE-LIKE TENSOR FIELD ON A SASAKIAN MANIFOLD

• Published : 2006.02.01
• 48 10

#### Abstract

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.

#### Keywords

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

#### References

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 https://doi.org/10.2748/tmj/1178242960
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 https://doi.org/10.2748/tmj/1178241341
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 https://doi.org/10.2307/1970165
13. Gr. Tsagas, On the spectrum of the Laplace operator for the exterior 2-forms, Tensor (N. S.) 33 (1979), no. 1, 94-96