Let M:D→V5 and Mˉ:D→Vˉ5 (D ⊂ R2) be two surfaces in the second order deformation in the Riemannian 5-spaces V5 and Vˉ5 of curvatures R, Rˉ respectively.
Let L: Tm (V5)→Tm (Vˉ5 ) be an isometry such that L(dm/dt) = dmˉ/dt'
𝝅: Tm(V5)→n = {V5}. Then M and Mˉ are in the third order deformation provided that: 1- The Gaussian curvature K and the curvature k of the normal bundle satisfies K2-k2 ≠ 0 on M. 2- dim T2m (M) = 4 on M.  3- M has no non-trivial real conjugate directions at each of its points  4- L {R(x,y)z} = Rˉ(Lx,Ly)Lz, and L{𝝅R(x,y)u} = 𝝅ˉRˉ(Lx,Ly)Lu for each x,y,z ϾTm(M), u Ͼ Nm(M) ={v3,v4}. 5- M and Mˉ are in the third order deformation on 𝜌 D.