The main purpose of the present paper is to study the geometry of Riemannian manifolds endowed with Golden structures. We discuss the problem of integrability for Golden Riemannian structures by using a φ-operator which is applied to pure tensor fields. | Turkish Journal of Mathematics Research Article Turk J Math (2013) 37: 693 – 703 ¨ ITAK ˙ c TUB doi: On integrability of Golden Riemannian structures ˙ Arif SALIMOV Aydın GEZER,∗ Nejmi CENGIZ, Ataturk University, Faculty of Science, Department of Mathematics, 25240 Erzurum, Turkey Received: • Accepted: • Published Online: • Printed: Abstract: The main purpose of the present paper is to study the geometry of Riemannian manifolds endowed with Golden structures. We discuss the problem of integrability for Golden Riemannian structures by using a φ -operator which is applied to pure tensor fields. Also, the curvature properties for Golden Riemannian metrics and some properties of twin Golden Riemannian metrics are investigated. Finally, some examples are presented. Key words: Golden structure, pure tensor, Riemannian manifold, twin metric 1. Introduction Let M be a C ∞ - manifold of finite dimension n. We denote by rs (M ) the module over F (M ) of all C ∞ -tensor fields of type (r, s) on M , . of contravariant degree r and covariant degree s, where F (M ) is the algebra of C ∞ -functions on M . Manifolds, tensor fields and connections are always assumed to be differentiable and of class C ∞ . Yano [25] introduced the notion of an f -structure which is a (1, 1)-tensor field of constant rank on M and satisfies the equality f 3 + f = 0 . This notion is a generalization of almost complex and almost contact structures. In its turn, it has been generalized by Goldberg and Yano [2], who defined a polynomial structure of degree d which is a (1, 1)-tensor field f of constant rank on M and satisfies the equation Q(f) = f d + ad f d−1 + . + a2 f + a1 I = 0 , where a1 , a2 , ., ad are real numbers and I is the identity tensor of type (1, 1). For a manifold M , let ϕ be a (1, 1)-tensor field on M . If the polynomial X 2 − X − 1 is the minimal polynomial for a structure ϕ .
