Novel theorems for metallic structures on the frame bundle of the second order

Mohammad Nazrul Islam Khana

Filomat, Tome 36 (2022) no. 13, p. 4471

Résumé

It is well known that ‘an almost complex structure’ J that is J2 = −I on the manifold M is called ‘an almost Hermitian manifold’ (M, J,G) if G(JX, JY) = G(X,Y) and proved that (F2M, JD,GD) is ‘an almost Hermitian manifold’ on the frame bundle of the second order F2M. The term ‘an almost complex structure’ refers to the general quadratic structure J2 = pJ + qI, where p = 0, q = −1. However, this paper aims to study the general quadratic equation J2 = pJ + qI, where p, q are positive integers, it is named as a metallic structure. The diagonal lift of the metallic structure J on the frame bundle of the second order F2M is studied and shows that it is also a metallic structure. The proposed theorem proves that the diagonal lift GD of a Riemannian metric G is a metallic Riemannian metric on F2M. Also, a new tensor field J˜ of type (1,1) is defined on F2M and proves that it is a metallic structure. The 2-form and its derivative dF of a tensor field J˜ are determined. Furthermore, the Nijenhuis tensor N J˜ of a metallic structure J˜ and the Nijenhuis tensor NJD of a tensor field JD of type (1,1) on the frame bundle of the second order F2M are calculated.

Détail

DOI : 10.2298/FIL2213471K

Classification : 53C15, 58D17
Keywords: Metallic structure, Nijenhuis tensor, Diagonal lift, Riemannian metric, Frame bundle