Novel theorems for the frame bundle endowed with metallic structures on an almost contact metric manifold

Abstract

It has been found that an almost complex structure of a contact metric manifold on frame bundle (FM,gD,J) is an almost Hermitian manifold. The derivative and coderivative of the Kähler form of the almost Hermitian structure (gD,J) are determined on frame bundle. An almost complex structure is a particular case of the polynomial structure of degree 2 satisfying J2=pJ+qI, where p=0,q=−1. However, the main contribution of this paper is that the results by applying the p,q as positive numbers then it satisfies the condition on J2=pJ+qI and termed as metallic structure. Furthermore, a tensor field J˜ is introduced on a frame bundle FM which proves that it is metallic structure on FM. The proposed theorem shows that the diagonal lift gD of a Riemannian metric g is a metallic Riemannian metric on FM. The derivative and coderivative of 2-form F of metallic Riemannian structure on FM are calculated. Moreover, the Nijenhuis tensor of tensor field J˜ is determined. Finally, a locally metallic Riemannian manifold (FM,JH,gD) is described as an application.