### On a Structure De ned by a Tensor Field F of Type (1, 1) Satisfying $\prod\limits_{j=1}^{k}$[F2+a(j)F+λ2(j)I]=0

DOI QR Code

Das, Lovejoy;Nivas, Ram;Singh, Abhishek

• 투고 : 2010.05.14
• 심사 : 2010.09.27
• 발행 : 2010.12.31
• 13 6

#### 초록

The differentiable manifold with f - structure were studied by many authors, for example: K. Yano [7], Ishihara [8], Das [4] among others but thus far we do not know the geometry of manifolds which are endowed with special polynomial $F_{a(j){\times}(j)$-structure satisfying $$\prod\limits_{j=1}^{k}\;[F^2+a(j)F+\lambda^2(j)I]\;=\;0$$ However, special quadratic structure manifold have been defined and studied by Sinha and Sharma [8]. The purpose of this paper is to study the geometry of differentiable manifolds equipped with such structures and define special polynomial structures for all values of j = 1, 2,$\ldots$,$K\;\in\;N$, and obtain integrability conditions of the distributions $\pi_m^j$ and ${\pi\limits^{\sim}}_m^j$.

#### 키워드

$F_{a(j),\lambda(j)}$-structure;distribution;integrability

#### 참고문헌

1. Lovejoy S. Das and Ram Nivas, On differentiable manifolds with [$F_1,\;F_2$](K + 1, 1) - structue, Tensor, N. S., 65(1) (2004), 29-35.
2. Lovejoy Das, Fiberings on almost r - contact manifolds, Publicationes Mathematicae, Debrecen, Hongrie, 43(1-2)(1993), 1-7.
3. Lovejoy Das, On CR - structures and F - structure satisfying $F^{K}+(-)^{K+1}F=0$, Rocky Mountain Journal of Mathematics, USA, 36(2006), 885-892. https://doi.org/10.1216/rmjm/1181069434
4. Lovejoy Das and Ram Nivas, Harmonic morphism on almost r - contact metric manifolds, Algebras Group and Geometries 22(2005), 61-68.
5. S. Helgason, Differential geometry, Lie groups and symmetric spaces, Academic Press, New York (1978).
6. S. Ishihara and K. Yano, On integrability conditions of a structure f satisfying $f^{3}+f=0$, Quart. J. Math., Oxford Sem (2) IS (1964), 217-222. https://doi.org/10.1093/qmath/15.1.217
7. R. S. Mishra, Structures on a Differentiable Manifold and their Applications, Chandrama Prakashan, 50-A, Balrampur House, Allahabad, India, 1984.
8. B. B. Sinha and R. Sharma, On a special quadratic structures on differentiable manifolds, Indian J. Pure Appl. Math., 9(8)(1978), 811-817.
9. K. Yano, On a structure defined by a tensor field f of type (1, 1) satisfying $f^{3}+f=0$, Tensor N. S., 14(1963), 99-109.