A class of $U(n)$-invariant Riemannian metrics on manifolds diffeomorphic to odd-dimensional spheres

It is well known that on the sphere $S^{2n-1}$ there is, up to multiplication by a scalar factor. a one-parameter family of $U(n)$-invariant metrics. We distinguish the class of normal metrics in the family, give precise estimates for sectional curvatures of the whole family of $U(n)$-invariant metrics, and distinguish metrics with small injectivity radius. Also, we prove local isometry of the three-dimensional spheres endowed with the metrics in question and the corresponding bundles of vectors of fixed length, over the two-dimensional spheres, endowed with the Sasakian metric.