On k-symmetries of hyperbolic spaces

We determine all the possible pointwise k-symmetric spaces of negative constant curvature. In general, such spaces are not k-symmetric. In fact we show that, for all n>= 3, k not 2, H^n is not k-symmetric, i.e., for any set of selected k-symmetries, one for each point of H^n, the regularity condition does not hold.