By Frohlich S.
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Additional resources for Surfaces in Euclidean spaces
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  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, .
Surfaces in Euclidean spaces by Frohlich S.