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 Apr21 asked Index of a zero of a normal vector field Apr20 awarded Nice Question Apr15 answered On the construction of hyperelliptic Riemann surfaces. Mar17 comment Line bundle of degree 1 on a genus 2 surface without global holomorphic sections This is in Gunning, Lectures on Riemann Surfaces, chapter 7.b, in the discussion after Theorem 13. Mar17 comment Line bundle of degree 1 on a genus 2 surface without global holomorphic sections Mar17 asked Line bundle of degree 1 on a genus 2 surface with 2 independent global holomorphic sections Mar9 asked Line bundle of degree 1 on a genus 2 surface without global holomorphic sections Mar9 answered Cocycle condition for line bundles. Mar2 accepted Divisor of meromorphic section of point bundle over a Riemann surface Feb22 accepted Length of geodesic representative on hyperbolic surfaces Feb22 comment Length of geodesic representative on hyperbolic surfaces yes; could you explain how it can be used to answer my question? Feb21 comment Length of geodesic representative on hyperbolic surfaces Thanks for this complete answer. There's a point I don't get. $\gamma_1 \delta$ is a path from $x$ to $y$, and $\alpha$ is the geodesic path which is path homotopic to $\gamma_1 \delta$, so I would expect it to be shorter than $\gamma_1 \delta$, i.e. $L(\alpha)\leq \dfrac{1}{2}L(\gamma)+L(\delta)$. But you wrote an inequality which is equivalent to $L(\delta)\leq \dfrac{1}{2}L(\gamma)+L(\alpha)$...how do you get it? Feb20 revised Hyperbolic (and related) structures on open unit disk deleted 11 characters in body Feb20 comment Dehn twist as isometries on hyperbolic surface I have edited my answer and tried to describe explicitly an example in genus 1. Hope it helps. Feb20 revised Dehn twist as isometries on hyperbolic surface added 2003 characters in body Feb16 revised Length of geodesic representative on hyperbolic surfaces added 11 characters in body Feb16 comment Length of geodesic representative on hyperbolic surfaces lower bound; something like $l(\gamma) > f(l(\alpha),l(\beta))$ for some function; I have edited the question, thanks Feb16 asked Length of geodesic representative on hyperbolic surfaces Feb16 asked Maximal tori in Lie vs algebraic groups Feb16 comment Maximal tori in lie groups? I think the statement is false, stated in this way. For instance in $\mathbb{C}^*$ a maximal torus (i.e. maximal compact connected abelian Lie subgroup) is given by $S^1$, which is not maximal abelian, since $\mathbb{C}^*$ itself is abelian. This of course generalizes to $GL(n,\mathbb{C})$ for higher $n$.