Metallic Quarter-Semi-Symmetric Non-Metric Connection

In this paper, we introduce the notion of the metallic quarter semi-symmetricnon-metric connection on metallic Riemannian manifolds. Constructed by deformingthe Levi-Civita connection via the metallic structure and two associated 1-forms, thisnew connection possesses several notable geometric properties. Crucially, we uncover astriking invariance property: the covariant derivative of the metallic structure is completely unaffected by this non-metric deformation. We establish the conditions underwhich its torsion tensor is cyclically parallel. Furthermore, we demonstrate that theNijenhuis tensor remains invariant, thereby preserving the integrability of the metallicstructure. We then derive the curvature and Ricci tensors, proving that if the manifold is locally metallic and the 1-forms are closed, the new curvature tensor coincidesexactly with that of the Levi-Civita connection. Extending our study to invariant submanifolds, we show they retain their totally geodesic, totally umbilical, and minimalproperties under this connection. Finally, we provide a non-trivial example in threedimensional Euclidean space to verify the existence and non-metric nature of this newlyintroduced connection.