The sphere $S^{2m-1} \simeq SU(m+1)/SU(m)$ has a canonical $U(1)$-action, and quotienting by this action give complex projective space $CP^m$. We can generalise the family of sphere to the family of spaces $S^{m,k}$ where we defines $$ S^{m,k} := SU(m+1)/(SU(k) \times SU(m-k)), ~~~~~~~~~~~~~~ k = 1, \ldots, m - 1. $$ It also have a $U(1)$-action, and quotienting by this give the Grassmanian $Gras(m,k)$.

{\bf What is correct the name and notation for $S^{m,k}$?}

The sphere $S^{2m-1} \simeq SU(m+1)/SU(m)$ has a canonical $U(1)$-action, and quotienting by this action give complex projective space $CP^m$. We can generalise the family of sphere to the family of spaces $S^{m,k}$ where we defines $$ S^{m,k} := SU(m+1)/(SU(k) \times SU(m-k)), ~~~~~~~~~~~~~~ k = 1, \ldots, m - 1. $$ It also have a $U(1)$-action, and quotienting by this give the Grassmanian $Gras(m,k)$.

What is correct the name and notation for $S^{m,k}$?