Dear members, If you don't mind I would like to ask the following question about the calculation of the injectivity radius on some space. Indeed, I almost determined the injectivity radius (local and global) of the space of matrices $diag(\lambda_{1},\lambda_{2},\ldots,\lambda_{m})V: \lambda_{1}\geq\lambda_{2}\geq\ldots\geq\lambda_{m}>0, \sum_{i=1}^{m}\lambda_i^2=1,V\in SO(m)$, is there any mean to deduce the injectivity radius of the space of matrices $diag(\lambda_{1},\lambda_{2},\ldots,\lambda_{m})V: \lambda_{1}>\lambda_{2}>\ldots>\lambda_{m}>0, \sum_{i=1}^{m}\lambda_i^2=1,V\in SO(m)$ Thank you for the help Riadh