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By Frohlich S.

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By Frohlich S.

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Let the immersion X : B → Rn+2 be given together with an ONF N. Let furthermore Lσ ϑ ,11 := 2 Lσ ,11 Lσ ,12 , W Lϑ ,11 Lϑ ,12 Lσ ϑ ,12 := 1 Lσ ,11 Lσ ,22 W Lϑ ,11 Lϑ ,22 Lσ ϑ ,22 := 2 Lσ ,12 Lσ ,22 , W Lϑ ,12 Lϑ ,22 for σ , ϑ = 1, . . , n. t. ,n with Kσ ϑ := Lσ ϑ ,11 Lσ ϑ ,22 − L2σ ϑ ,12 W2 for σ , ϑ = 1, . . , n. The curvatures Hσ ϑ and Kσ ϑ are sectional curvatures in the following sense. 5. For all σ , ϑ = 1, . . , n there hold the following statements. 1. t. parameter transformations of class P.

We want to refer the reader to da Costa [37] for an application in quantum mechanics in curved spaces. 10 The fundamental theorem We want to reconstruct an immersion X from given first and second fundamental forms, given torsion coefficients, and a given (n + 2)-frame attached at some point of the surface, say at (0, 0) ∈ B. The latter assumption is needed to construct an initial (n + 2)-frame at an arbitrary point of the surface. In particular, it is needed in the second point of our proof below: Assume we know that the tangential planes at a certain point w ∈ B of two solutions X and X coincide, and therefore the normal spaces are the same.

25) we arrive at the linear system (1) fi j,uk = (2) fiσ ,uk = 2 2 ∑ Γikm fm j + ∑ Γjkm fmi (1) (1) 2 ∑ Γikm fmσ + Lσ ,ik fσ (2) (3) m=1 n ∑ Tσϑ,k fiϑ (2) ∑ Lϑ ,ik f jϑ 2 − (2) n + ϑ =1 m=1 m=1 + n + ∑ ∑ Lϑ , jk fiϑ (2) , ϑ =1 (1) Lσ ,km gmn fin + m,n=1 n ∑ Lϑ ,ik fσ ϑ (4) ϑ =1 σ =ϑ , ϑ =1 2 ∑ (3) fσ ,uk = − 2 m,n=1 2 (4) fσ ϑ ,uk = − ∑ m,n=1 n + (2) Lσ ,km gmn fnσ + (2) Lσ ,km gmn fnϑ − n ∑ Tσϑ,k fσ ϑ , (4) ϑ =1 σ =ϑ 2 ∑ m,n=1 (2) Lϑ ,km gmn fnσ + n ∑ Tσω,k fωϑ (4) ω =1 ω =ϑ ∑ Tϑω,k fωσ (4) ω =1 ω =ϑ with the same initial conditions as the trivial solution 0, .

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Surfaces in Euclidean spaces by Frohlich S.


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