A RIGIDITY THEOREM FOR SURFACES IN RIEMANNIAN 3-SPACES. | ||||
| The International Conference on Applied Mechanics and Mechanical Engineering | ||||
| Article 54, Volume 1, 2nd Conference on Applied Mechanical Engineering., May 1986, Page 15-23 PDF (1.57 MB) | ||||
| Document Type: Original Article | ||||
| DOI: 10.21608/amme.1986.52561 | ||||
 View on SCiNiTO | ||||
| Author | ||||
| RAMY M.K. TALAAT | ||||
| Gen. Dr., Military Technical College, Kobry El-Kobba, Cairo, Egypt. | ||||
| Abstract | ||||
| 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. | ||||
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