Naturality of homogeneous metrics on Stiefel manifolds SO(m+1)/SO(m−1)SO(m+1)/SO(m−1)

It is well known that the unit tangent sphere bundle T1SmT1Sm of the standard sphere SmSm can be naturally identified with the Stiefel manifold V2Rm+1=SO(m+1)/SO(m−1)V2Rm+1=SO(m+1)/SO(m−1). In this paper, we construct the (1–1)(1–1) correspondence between all SO(m+1)SO(m+1)-invariant homogeneous metrics on V2Rm+1V2Rm+1 and all so-called g  -natural metrics on T1SmT1Sm.