A class of \(U(n)\)-invariant Riemannian metrics on manifolds diffeomorphic to odd-dimensional spheres
The authors study \(U(n)\)-invariant Riemannian metrics on the sphere \(S^{2n - 1} = U(n) / U(n - 1)\). Such metrics \(g\) up to a homothety depend on one parameter. The sectional curvature of \(g\) is calculated and the radius of injectivity is estimated. The authors show that some of these invariant metrics \(g\) on an odd dimensional sphere have a positive sectional curvature \(K\), \(0 < K \leq \lambda\), but the radius of injectivity \(d_ p\) in any point \(p\) does not satisfy Klingenberg's inequality \(d_ p \geq \pi/\sqrt{\lambda}\) which is valid for positively curved even dimensional manifolds. It is proved also that the sphere \(S^ 3\) with the standard metric of sectional curvature 1 is locally isometric to the sphere bundle of tangent vectors of length 1/2 on the 2- sphere \(S^ 2\) of radius 1/2 equipped with the Sasakian metric.
scientific article; zbMATH DE number 652168