Kurver og flader Aktivitet 15 Geodætiske kurver, Isometri, Mainardi-Codazzi, Teorema Egregium

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1 Kurver og flader Aktivitet 15 Geodætiske kurver, Isometri, Mainardi-Codazzi, Teorema Egregium Lisbeth Fajstrup Institut for Matematiske Fag Aalborg Universitet Kurver og Flader 2013 Lisbeth Fajstrup (AAU) Kurver og flader 2013 November / 9

2 Proposition A curve γ on a surface S is a geodesic if and only if, for any part γ(t) = σ(u(t), v(t)) of γ contained in a surface patch σ of S, the following two equations are satisfied: OBS: ü + Γ 1 11 ( u)2 + 2Γ 1 12 u v + Γ1 22 ( v)2 = 0 v + Γ 2 11 ( u)2 + 2Γ 2 12 u v + Γ2 22 ( v)2 = 0 Γ k ij are functions of t as follows: Γ k ij (u(t), v(t)). Γ k ij (u, v) are functions of E(u, v), F(u, v), G(u, v) and the derivatives of these. Lisbeth Fajstrup (AAU) Kurver og flader 2013 November / 9

3 Thm A curve γ on a surface S is a geodesic if and only if, for any part γ(t) = σ(u(t), v(t)) of γ contained in a surface patch σ of S, the following two equations are satisfied: d dt (E u + F v) = 1 2 (E u u 2 + 2F u u v + G u v 2 ) d dt (F u + G v) = 1 2 (E v u 2 + 2F v u v + G v v 2 ) Lisbeth Fajstrup (AAU) Kurver og flader 2013 November / 9

4 Corollary A local diffeomorphism f : S 1 S 2 is a local isometry if and only if, for any surface patch, σ : U S 1 such that σ 2 = f σ : U S 2 is a surface patch, σ and σ 2 have the same first fundamental form. In slogan form: If you know E(u, v), F(u, v), G(u, v), then you know the surface up to local isometry of surfaces. Corollary Geodesic curves are preserved under local isometry. Lisbeth Fajstrup (AAU) Kurver og flader 2013 November / 9

5 Proposition (Gauss equations) Let K (σ(u, v)) be the Gaussian curvature of a surface expressed in terms of a surface patch σ, then FK = (Γ 1 12 ) u Γ 1 11 ) v + Γ 1 12 Γ2 12 Γ2 11 Γ1 22 (and there are three similar expressions in [AP]) Teorema Egregium Gaussian Curvature is preserved under local isometry. Principal curvature κ 1,, κ 2 are not preserved under isometry. Mean curvature 1 2 κ 1 + κ 2 is not preserved under isometry. BUT K = κ 1 κ 2 is preserved. Lisbeth Fajstrup (AAU) Kurver og flader 2013 November / 9

6 Teorema Egregium og landkort Sætning: Der findes ikke noget landkort, der har konstant målestok. Bevis: Det ville kunne skaleres til en isometri. Sætning: Der findes ikke noget landkort, der bevarer både vinkler og areal. Vinkelbevarende OG arealbevarende isometri (Sætning (Korollar 6.3.4) og sætning 6.4.5) Den gode nyhed Der findes landkort, der bevarer vinkler (e.g. Mercator projektion, stereografisk projektion) Der findes landkort, der bevarer areal (e.g. Archimedes projektion, Peters projektionen) Der findes landkort, der sender storcirkler i storcirkler (men så bevarer det ikke vinkler) Lisbeth Fajstrup (AAU) Kurver og flader 2013 November / 9

7 Proposition (Codazzi Mainardi) Let E, F, G, L, M, N be the coefficients of the first and second fundamental form of a surface patch σ(u, v). Then L v M u = LΓ M(Γ2 12 Γ1 11 ) NΓ2 11 M v N u = LΓ M(Γ2 22 Γ1 12 ) NΓ2 12 Lisbeth Fajstrup (AAU) Kurver og flader 2013 November / 9

8 Theorem Let σ 1, σ 2 : U R 3 be surface patches with the same first and second fundamental form. Then there is a direct isometry M : R 3 R 3, M(x) = Ax + b. A SO(3), such that σ 2 = M σ 1. (uniqueness) Suppose E, F, G, L, M, N : V R smooth, V R 2, E > 0, G > 0, EG F 2 > 0 E, F, G, L, M, N satisfy the Mainardi Codazzi equations The Gauss equations hold for K (u, v) = LN M2 EG F 2 Then, for any p V, there is a neighborhood p U V and a surface pathc defined on U such that E, F, G, L, M, N : U R are the coefficients in the first and second fundamental form for σ (existence of solution to certain partial differential equations) Slogan form of the uniqueness part: If you know E, F, G, L, N, M(u, v), then you know the surface up to isometry of the ambient space R 3. Lisbeth Fajstrup (AAU) Kurver og flader 2013 November / 9

9 Ideas and caveats in the proofs (of Teorema Egregium and Mainardi Codazzi). Write out the equation (σ uu ) v = (σ uv ) u in the basis (σ u, σ v, N). This gives equations for the coefficients of the basic vectors. Be careful when differentiating - how does the chain rule apply, e.g. Nu means u N(σ(u, v)). The chain rule, and the definition of de differential implies N u = D p G(σ u ) T G(p) S 2 = T p S (OBS: The last equation is crucial - see p. 163) Write it out, if you are not sure, what the short form means. N u = D p G(σ u ) is short for N u (σ(u, v)) = D σ(u,v) G(σ u (u, v)) Now use that Wp is the matrix of D p G wrt. the basis (σ u, σ v ). And that ( ) 1 ( ) E F L M W p = F G M N Lisbeth Fajstrup (AAU) Kurver og flader 2013 November / 9