The topological aspect of the holonomy displacement on the principal U(n) bundles over Grassmannian manifolds

Consider the principal U(n) bundles over Grassmann manifolds U(n)→U(n+m)/U(m)→πGn,m. Given X∈Um,n(C) and a 2-dimensional subspace m′⊂m ⊂u(m+n), assume either m′ is induced by X,Y∈Um,n(C) with X⁎Y=μIn for some μ∈R or by X,iX∈Um,n(C). Then m′ gives rise to a complete totally geodesic surface S in the base space. Furthermore, let γ be a piecewise smooth, simple closed curve on S parametrized by 0≤t≤1, and γ˜ be its horizontal lift on the bundle U(n)→π−1(S)→πS, which is immersed in U(n)→U(n+m)/U(m)→πGn,m. Thenγ˜(1)=γ˜(0)⋅(eiθIn)orγ˜(1)=γ˜(0), depending on whether the immersed bundle is flat or not, where A(γ) is the area of the region on the surface S surrounded by γ and θ=2⋅n+m2nA(γ).