Higher order symmetric spaces and the roots of the identity in a Lie group

Let r(k) (G) denote the set of all k-roots of the identity in a Lie group G . We show that r(k)(G) is always an embedded submanifold of G, having the c onjugacy classes of its elements as open submanifolds. These conjugacy clas ses are examples of k-symmetric spaces and we show, more generally, that ev ery k-symmetric space of a Lie group G is a covering manifold of an embedde d submanifold Orb of G. We compute also the Hessian of the inclusions of r( k)(G) and Orb into G, relative to the natural connection on the domain and to the symmetric connection on G.