METALLIC QUARTER-SEMI-SYMMETRIC NON-METRIC CONNECTION

 In this paper, we introduce the notion of the metallic quarter semi-symmetric
non-metric connection on metallic Riemannian manifolds. Constructed by deforming
the Levi-Civita connection via the metallic structure and two associated 1-forms, this
new connection possesses several notable geometric properties. Crucially, we uncover a
striking invariance property: the covariant derivative of the metallic structure is com
pletely unaffected by this non-metric deformation. We establish the conditions under
which its torsion tensor is cyclically parallel. Furthermore, we demonstrate that the
Nijenhuis tensor remains invariant, thereby preserving the integrability of the metallic
structure. We then derive the curvature and Ricci tensors, proving that if the mani
fold is locally metallic and the 1-forms are closed, the new curvature tensor coincides
exactly with that of the Levi-Civita connection. Extending our study to invariant sub
manifolds, we show they retain their totally geodesic, totally umbilical, and minimal
properties under this connection. Finally, we provide a non-trivial example in three
dimensional Euclidean space to verify the existence and non-metric nature of this newly
introduced connection.