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## New answers tagged riemann-surfaces

1

What you want is the reason for naming a theorem this way. So the answer has to be some kind of guess and by analogy. It is possible in algebraic topology to find two non-homeomorphic topological spaces having the same fundamental groups (or homology groups). That is within a "collection of topological spaces" of same fundamental groups one can find two ...

0

Here is a solution in great generality: Let $X$ be a Riemann surface, $Y$ its universal cover, and $\pi:Y\rightarrow X$ the universal covering map. Let $\Sigma=\big\{\{U_i\},\{\varphi_j\}\big\}$ be the complex structure on $X$. Then, $\Sigma$ may be lifted to $Y$ by way of $\pi$ to induce a complex structure $\Sigma'=\big\{\{\pi^{-1}|_{U_i}\},\{\psi_j\... 0 This is an addendum to the question: There exists a 2-generated subgroup$\Gamma$of isometries of the hyperbolic 5-space${\mathbb H}^5$such that every element of$\Gamma$is elliptic but$\Gamma$does not fix a point in${\mathbb H}^5$. (An isometry of a space$X$of curvature$\le 0$is elliptic when it fixes a point in$X$. I will use this ... 0 Let $$H_n = \{ w_1 \frac{a}{n}+w_2\frac{b}{n}, a,b \in 0 \ldots n-1, 0 < a+b\}$$ And for$n$odd $$h_n = \{ w_1 \frac{a}{n}+w_2\frac{b}{n}, a,b \in 0 \ldots n-1,0 < a+b< n\}$$ Since$\wp(u) = \wp(-u)$and$H_n = h_n \cup -h_n \bmod \Lambda$(disjoint union) we have $$\prod_{u \in H_n} (\wp(z)-\wp(u))=(\prod_{u \in h_n} (\wp(z)-\wp(u)))^2 =P_n(\... 2 Unfortunately You can't just use D/\Gamma because then the boundary S^1 will be very badly behaved (e.g., nonexistence of nontangential limit by choosing the appropriate point on each fundamental 4g-gon). The usual introductory way is to just use Hilbert space theory with minimal amount of Sobolev spaces thrown in. (I+\Delta)^{-1}\colon L^2(M)\to H^2(... 1 They are sometimes called cone points, because they look like taking a piece of paper with a corner of angle 2\pi/m_i and folding it over to identify opposite sides of the vertex. This produces something which looks like a cone at the singular point. They are also sometimes called pillowcase points as they look like the corners of a pillow. So, for ... 1 This follows straight from the transformations that have been developed, namely$$\left< \hat{e}_i, \hat{e}^{*j}\right> = (\sum_k g_{ik}\hat{e}^{*k}) \cdot \hat{e}^{*j} = \sum_k g_{ik}g^{kj} = \delta_i^j = \begin{cases} 0 \ i\neq j\\ 1 \ i = j \end{cases}$$As the author states in the text, that the basis$\{ e_i \}$or$\{ e^{i*} \}$may not be ... 2 Let me first consider the case of nonconstant holomorphic maps$X\to T^2$, from compact connected Riemann surfaces of genus$g\ge 2$to tori (smooth elliptic curves). Every such map is determined (up to a translation of$T^2$) by the induced map of fundamental groups$G=\pi_1(X)\to \pi_1(T^2)=Z^2$. There are only countably many homomorphisms$G\to Z^2\$ and ...

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