For a constrained minimization problem, the restriction of the Hessian of the Lagrangian to a tangent space of the feasible set can be used to characterize a Karush-Kuhn-Tucker point of the problem as a local minimum, maximum or saddle point. It is shown in this paper that the restriction of the Hessian to a normal space with respect to the indefinite inner product it induces can be used to characterize a Karush- Kuhn-Tucker point for the Wolfe dual problem. Another result is that, under a regularity condition, the Hessian is positive semidefinite if and only if the considered Karush-Kuhn-Tucker point satisfies the second order necessary condition for a local minimum of the primal problem and, at the same time, satisfies the second order necessary condition for a local maximum of the dual.