Sbe an oriented closed surface of genus exceeding one and ‰: … 1(S)!SL(2;C ) a nonelementary representation. Then ‰is the monodromy of a holomorphic °at connection on a maximally unstable holomorphic vector bundle of rank two over a Riemann surface R, where Ris diﬁeomorphic to Svia an orientation preserving diﬁeomorphism R!S.

In mathematics, a dessin d'enfant is a type of graph embedding used to study Riemann surfaces and to provide combinatorial invariants for the action of the absolute Galois group of the rational numbers.

G.2. Branch Point and Riemann Surface This section provides background knowledge for stability analysis based on Nyquist method. Definition of branch point If a function f(z) has a branch point at z=z 0, then f(z 0+εe iθ) is different from f(z 0+εe i(θ+2π)). Definition of Riemann surface Riemann surface is a surface of z=z 0+εe iθ where ... The integrability conditions for spinor transforms lead naturally to Dirac spinors and their application to conformal immersions. The second part presents a complete spinor calculus on a Riemann surface, the definition of a conformal Dirac operator, and a generalized Weierstrass representation valid for all surfaces. The printer’s sheet (pechatnyi list) is used to calculate the actual length of a publication; the unit of measure is a paper sheet 60 × 90 cm, printed on one side. For paper of other dimensions (for example, 70 × 90 cm), the term “actual printer’s sheet” is used. z=infinity is not branch point. Valid branch cuts: z−1=r 1 e iθ1 z+1=r2 e iθ2 f(z)=(r1 r2) 1/2 e i( θ1+ θ2. • Riemann surface is a closed, 2-sheeted surface such that any cycle surrounding one of the branch points brings us to a new sheet, whereas any cycle surrounding both branch points restores f to initial value. This is because the double of any two triangles is the same, as a Riemann surface. 3 Sheaves and analytic continuation. Presheaves and sheaves. A presheaf of abelian groups on Xis a functor F(U) from the category of open sets in X, with inclusions, to the category of abelian groups, with homomorphisms.

geodesic on a Riemann surface M with length l. Then there is a holomorphic metric-preserving mapping ϕ: T → M such that ϕ(iy)=γ. The image ϕ(T) of T is called a collar. Then we can choose θ 0 smallenoughsuchthattheareaofthe Collarisatleast √8 5. Translating the above theorem into Riemannian geometry, we let C R be the collar. By Keen’s theorem, C