On Kähler–Norden–Codazzi golden structures on pseudo-Riemannian manifolds

Abstract

In this paper, we consider a pseudo-Riemannian manifold equipped with a Kähler–Norden–Codazzi golden structure. For such a manifold, we study curvature properties. Also, we define special connections of the first type and of the second type on the manifold, which preserve the associated twin Norden golden metric and satisfy some special conditions and present some results concerning them.

Some notes on metallic Kähler manifolds

“…Almost complex golden structures were defined in [2]. Note that there is a bijection between almost complex structures and almost complex golden structure, as it is shown in [1,3]. If a (1, 1)−tensor field J provides the equation x 2 − px + 3 2 q = 0, then we call it an almost complex metallic structure.…”

Classification of Almost Norden Golden Manifolds

“…These numbers explain the name of both kind of polynomial structures. Since then, almost golden and almost complex golden structures have become in active research field (see, e.g., [1,3,6,10,11] and the references therein).…”

“…In [1,11], the authors also studied almost complex golden structures which admit compatible pseudo-Riemannian metrics. Compatible metrics on almost complex golden manifolds are introduced in the same way that Norden metrics on almost complex manifolds.…”

A Neutral relation between metallic structure and almost quadratic φ -structure

“…where J is a (1, 1) tensor field on M, I is the identity operator on the Lie algebra Γ(T M ) of vector fields on M and p, q are real numbers. This structure can be also viewed as a generalization of following well known structures : · If p = 0, q = 1, then J is called almost product or almost para complex structure and denoted by F [15], [12], · If p = 0, q = −1, then J is called almost complex structure [17], · If p = 1, q = 1, then J is called golden structure [6], [7], · If p is positive integer and q = −1, then J is called poly-Norden structure [16], · If p = 1, q = −3 2 , then J is called almost complex golden structure [3], · If p and q are positive integers, then J is called metallic structure [11]. If a differentiable manifold endowed with a metallic structure J then the pair (M, J) is called metallic manifold.…”

“…• If p = −1, q = 3 2 , then J is called an almost complex golden structure [1]; • If p and q are positive integers, then J is called a metallic structure [11].…”