# Surfaces in Euclidean Spaces by Steffen Fröhlich By Steffen Fröhlich

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N. 2 µ ∑ Tσϑ,µ Λi µ =1 32 2 Differential equations. 7 Weingarten forms. Principal curvatures In this section we want to introduce the so-called Weingarten forms to define algebraically the principal curvatures of an immersion. 8. t. some unit normal vector N ∈ Rn+2 are defined by LkN,i := LN,i j g jk for i, k = 1, 2. 9) along the unit normal vector N we infer L1N,1 + L2N,2 = 2 2 j=1 j=1 ∑ LN,1 j g j1 + ∑ LN,2 j g j2 2 = ∑ LN,i j g ji = 2HN i, j=1 as well as 2 L1N,1 L2N,2 − (L2N,1 ) = ∑ i, j=1 LN,1i LN,2 j gi1 g j2 − 2 ∑ LN,1i gi2 LN,2 j g j1 i, j=1 = (LN,11 LN,22 − L2N,12 )(g11 g22 − g12g21 ) = KN .

2 2 j,k=1 j,k=1 ∑ (L2 j Lk1 − L1 j Lk2 )g jk = ∑ (L2 j L j1 − L1 j Lk2 )g jk = 0. 12) invites us to define a curvature of the normal bundle analogously to our definition of the Riemannian curvature tensor in terms of the Christoffel symbols Rℓi jk = ∂uk Γi ℓj − ∂u j Γikℓ + 2 ∑ (Γi mj Γmkℓ − ΓikmΓmℓj ). m=1 46 3 Integrability conditions The normal space of a surface at w ∈ B was introduced as NX (w) = Z ∈ Rn+2 : Z · Xu (w) = Z · Xv (w) = 0 . 2. 13) mn for i, j = 1, 2 and σ , ω = 1, . . , n. 13) is due to the Ricci equations.

This leads us to a central notion of our investigations. 3. The normal bundle NX is called flat if and only there hold Sσω,12 ≡ 0 for all σ , ω = 1, . . t. to some ONF N. Now we come to the proof of the foregoing proposition. Proof. We introduce conformal parameter (u, v) ∈ B. Let again Nσ = cos ϕ Nσ + sin ϕ Nω , Nω = − sin ϕ Nσ + cos ϕ Nω , and insert it into the representation of Sσω,12 using Ricci’s integrability conditions. 48 3 Integrability conditions Then we compute W Sσω,12 = (Lσ ,11 Lω ,12 − Lσ ,21 Lω ,11 ) + (Lσ ,12Lω ,22 − Lσ ,22Lω ,21 ) = (cos ϕ Lσ ,11 + sin ϕ Lω ,11 )(− sin ϕ Lσ ,12 + cos ϕ Lω ,12 ) − (cos ϕ Lσ ,21 + sin ϕ Lω ,21 )(− sin ϕ Lσ ,11 + cos ϕ Lω ,11 ) + (cos ϕ Lσ ,12 + sin ϕ Lω ,12 )(− sin ϕ Lσ ,22 + cos ϕ Lω ,22 ) − (cos ϕ Lσ ,22 + sin ϕ Lω ,22 )(− sin ϕ Lσ ,21 + cos ϕ Lω ,21 ) = (Lσ ,11 − Lσ ,22 )Lω ,12 − (Lω ,11 − Lω ,22 )Lσ ,12 = W Sσω,12 , which proves the statement.