Complex Analytic Sets (Mathematics and its Applications) by E.M. Chirka

By E.M. Chirka

One provider arithmetic has rendered the 'Et moi, .. " si j'avait so remark en revenir, human race. It has positioned logic again je n'y semis aspect aile.' Jules Verne the place it belongs, at the topmost shelf subsequent to the dusty canister labelled 'discarded non­ The sequence is divergent; as a result we might be sense'. capable of do anything with it Eric T. Bell o. Heaviside arithmetic is a device for suggestion. A hugely important software in an international the place either suggestions and non­ linearities abound. equally, every kind of elements of arithmetic function instruments for different elements and for different sciences. using an easy rewriting rule to the quote at the correct above one unearths such statements as: 'One carrier topology has rendered mathematical physics .. .'; 'One carrier common sense has rendered com­ puter technology .. .'; 'One carrier classification concept has rendered arithmetic .. .'. All arguably precise. And all statements available this fashion shape a part of the raison d'etre of this sequence.

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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.

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