Affiliation:
1. Department of Computer Engineering, College of Computer, Qassim University, Buraydah, Saudi Arabia
Abstract
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 F2Mis 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.
Publisher
National Library of Serbia
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