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.
Talaat, R. (1986). DEFORMATION OF SURFACES IN RIEMANNIAN 5-SPACES. The International Conference on Applied Mechanics and Mechanical Engineering, 2(2nd Conference on Applied Mechanical Engineering.), 117-129. doi: 10.21608/amme.1986.57231
MLA
Ramy M.K. Talaat. "DEFORMATION OF SURFACES IN RIEMANNIAN 5-SPACES", The International Conference on Applied Mechanics and Mechanical Engineering, 2, 2nd Conference on Applied Mechanical Engineering., 1986, 117-129. doi: 10.21608/amme.1986.57231
HARVARD
Talaat, R. (1986). 'DEFORMATION OF SURFACES IN RIEMANNIAN 5-SPACES', The International Conference on Applied Mechanics and Mechanical Engineering, 2(2nd Conference on Applied Mechanical Engineering.), pp. 117-129. doi: 10.21608/amme.1986.57231
VANCOUVER
Talaat, R. DEFORMATION OF SURFACES IN RIEMANNIAN 5-SPACES. The International Conference on Applied Mechanics and Mechanical Engineering, 1986; 2(2nd Conference on Applied Mechanical Engineering.): 117-129. doi: 10.21608/amme.1986.57231
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