Hi, Consider P the space of n by n positive definite matrices. Let <X,Y>_p = trace(p^-1 X p^-1 Y) where p in P be the metric tensor on P so that it is a Riemannian manifold. The general linear group G acts on P by phi: G X P -> P, phi(g,p) = phi_g(p)=gpg' (g' means g transpose). This a transitive action by isometries. Does this constitute the isometry group on P however, that is, can every isometry on P be written in this fashion? Thanks!