Liftings of metallic structures to tangent bundles of order r

Abstract

It is well known that the prolongation of an almost complex structure from a manifold $ M $ to the tangent bundle of order $ r $ on $ M $ is also an almost complex structure if it is integrable. The general quadratic structure $ F^2 = \alpha F+\beta I $ is a generalization of an almost complex structure where $ \alpha = 0, \; \beta = -1. $ The purpose of this paper is to characterize a metallic structure defined by the general quadratic structure $ F^2 = \alpha F+\beta I, \; \alpha, \beta\in \mathbb{N} $, where $ \mathbb{N} $ is the set of natural numbers. We show that the $ r $-lift of the metallic structure $ F $ in the tangent bundle of order $ r $ is also a metallic structure. Furthermore, we deduce a theorem on the projection tensor in the tangent bundle of order $ r $. Moreover, prolongations of $ G $-structures immersed in the metallic structure to the tangent bundle of order $ r $ and 2 are discussed. Finally, we construct examples of metallic structures that admit an almost para contact structure on the tangent bundle of order 3 and 4.

Keywords

• lifts
• metallic structure
• integrability
• projection tensor
• prolongation