We study the isometries of an n-dimensional curved space S by studying linear transformations of an n+1 dimensional vector space V, with a quadratic form Q, over the reals.
S is composed of those points in V where the Q = 1 and thus these transformations map S to itself.
If we take the quadratic form as a metric for the space then these transformations are isometries of V, a flat space, but simple translations of V are also isometries of V.