The Geodesic Gauss Map and Ruh-Vilms theorem for a Hypersurface in S^{n}

We are interested to work on normal homogeneous space and in this space we calculated Live-Civita connection and we derived a useful equation 2.17. The Ruh-Vilms theorem is a statement about the Gauss map for a submanifold of R^{n+1} . Our aim is to prove, an isometrically immersed hypersurface f : M −→ S^{n} has constant mean curvature if and only if the Gauss map of γ is harmonic. Here we provide a proof of the Ruh-Vilms result using Homogeneous geometry. First shown for curves in S^{2} , then proven for a hypersurface in the n-sphere by using symmetric space identification and results in 2.17