dg.differential geometry – Map of Gram Schmidt as partial isometry

We equip $ Gl (n, mathbb {R}) $ Y $ O (n) $ with its invariant metrics on the left whose restrictions on the corresponding neutral elements is the standard internal product $ tr (ABtr) $ of $ M_n ( mathbb {R}) $.

Leave $ f: Gl (n, mathbb {R}) a O (n) $ I know the map of Gram Schmidt.

Is $ f $ A partial isometry with respect to these Riemannian structures? If the answer is negative, can we change the Riemannian metrics of these spaces (not necessarily invariable with respect to group operations) so that they have $ f $ As a partial isometry?