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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,...
Consider the Lie group SO0(n, 1) with the left-invariant metric coming from the Killing-Cartan form. The maximal compact subgroup SO(n) of the isometry group acts from the left and right. This paper studies the geometry of the quotient space of the homogeneous submersion SO0(n, 1) → SO(n)\SO0(n, 1). It is a cohomogeneity one manifold, which can be expressed as a warped product. Its group of isometries,...
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