Naturality of homogeneous metrics on stiefel manifolds \(SO(m + 1)/SO(m - 1)\)
The sphere tangent bundle \(T_1S^m\) of the standard sphere \(S^m\) with the natural action of the group \(SO(m+1))\) is a homogeneous manifold \(SO(m+1)/SO(m-1)\) and it is naturally identified with the Stiefel manifold \(V_2(\mathbb{R}^{m+1}) = SO(m+1)/SO(m-1) \) of 2-frames. The authors give a survey of results about different extensions of a Riemannian metric of a manifold \(M\) to the tangent bundle \(TM\) and the sphere tangent bundle \(T_1(M)\). They prove that so called \(g\)-natural metrics on \(T_1S^m\) correspond exactly to \(SO(m+1)\)-invariant metrics on the Stiefel manifold \(V_2(\mathbb{R}^{m+1})\).
scientific article; zbMATH DE number 5687508
scientific article