On a Study of the Totally Umbilical Semi-Invariant Submanifolds of Golden Riemannian Manifolds

Abstract

The Golden Ratio is fascinating topic that continually generated news ideas. A Riemannian manifold endowed with a Golden Structure will be called a Golden Riemannian manifold. Precisely, we can say that an (1,1)-tensor field (P) over bar on a m-dimensional Riemann manifold (<(M,<(g)over bar>)over bar>) is a Golden structure if it satisfies the equation (P) over bar (2) = (P) over bar + Id, where Id is identity map on M. Furthermore, g((P) over bar, (X) over bar, (Y) over bar) = (g) over bar((X) over bar,(P) over bar (Y) over bar), the Riemannian metric is called (P) over bar -compatible and ((M) over bar,(g) over bar,(P) over bar) is named a Golden Riemannian manifold. The main purpose of the present paper is to study the geometry of Riemannian manifolds endowed with Golden structures. For this purpose, we study totally umbilical semi-invariant submanifold of the Golden Riemannian manifolds. Also, we obtain integrability conditions of the distributions and investigate the geometry of foliations.