Let M : D→V³ and M→V-³ (D c R²) be two isometric surfaces in the Riemannian spaces V³ and V-³ with curvatures R, R- respectively.  We shall prove that the second fundamental forms of the two surfaces are the same provided that:
1- The Gaussian curvature K of M is positive.
2- M and M- have the same second fundamental form on ρ D.
3- For each d ϵ D, Ld :T M(d) (V³)→T M-(d)(V-³ ) is the isometry determined by its restriction Ld to T M (d) (M) which satisfies LdodM = dM-, and Ld {R(x,y) Z} 
= R-(Ldx, Ldy)Ldz for all tangent vectors x,y,z ϵ T M(d)(M)
Also it is shown that the two isometric surfaces M and M- satisfying the above conditions have the same Gaussian and mean curvatures at corresponding points.