International Journal of Mathematics and Computer Applications Research (IJMCAR) ISSN(P): 2249-6955; ISSN(E): 2249-8060 Vol. 5, Issue 6, Dec 2015, 37-44 © TJPRC Pvt. Ltd
FLAT H-CURVATURE TENSORS ON HSU-STRUCTURE MANIFOLD LATA BISHT1 & SANDHANA SHANKER2 1 2
Department of Applied Science, BTKIT, Dwarahat, Almora, Uttarakhand, India
Assistant Professor, Department of Mathematics, REVA University, Bangalore, Karnataka, India
ABSTRACT The main object of this paper is to study the flatness of the Bochner flat, H-Projectively flat, HConharmonically flat, H-Concircurally flat, Conharmonically* flat and Conformally* flat Hsu-structure manifold. KEYWORDS: Riemannian Curvature Tensor, Bochner Curvature Tensor, H-Projective Curvature Tensor, H- ConHarmonic Curvature Tensor, H-Con-Circular Curvature Tensor, Conharmonic* Curvature Tensor, Conformal* Curvature Tensor and Hsu-Structure Manifold
Received: Nov 10, 2015; Accepted: Nov 16, 2015; Published: Nov 21, 2015; Paper Id.: IJMCARDEC20154
1. INTRODUCTION If on an even dimensional manifold Vn, n = 2m of differentiability class C∞, there exists a vector valued real linear function
φ , satisfying
φ 2 = ar I n ,
(1.1a)
φ 2 X = a r X , for arbitrary vector field X.
(1.1b)
where X = φX , 0 ≤ r ≤ n and ' a ' is a real or imaginary number. Then { φ } is said to give to Vn a Hsu-structure defined by the equations (1.1) and the manifold Vn is called a Hsu-structure manifold. Remark (1.1): The equation (1.1) a gives different structure for different values of ' a ' and 'r'. If r = 0, it is an almost product structure, if a = 0, it is an almost tangent structure, if r = ±1 and a = +1, it is an almost product structure, if r = ±1 and a = −1, it is an almost complex structure, if
r = 2 then it is a GF-structure
which includes π-structure for a ≠ 0, an almost complex structure for a = ±i, an almost product structure for a = ±1, an almost tangent structure for a =0. Let the Hsu-structure be endowed with a metric tensor g, such that
g (φX ,φY ) + a r g ( X , Y ) = 0 . Then {φ, g} is said to give to Vn - metric Hsu-structure and Vn is called a metric Hsu-structure manifold. Agreement(1.1): In what follows and the above, the equations containing X,Y,Z., etc. hold for these arbitrary vector in Vn. The curvature tensor K, a vector -valued tri-linear function w.r.t. the connexion D is given by
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