Let S be a hyperbolic surface. Let p:M ->S be a covering.
My question: Is there a metric on M such that M is a hyperbolic surface and p is a local isometry?
Metric on a covering space of a hyperbolic surface
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differential-geometry
hyperbolic-geometry
1 Answers
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Let $x\in M, u,v\in T_xM$ define $\langle u,v\rangle_x=\langle dp_x(u),dp_x(v)\rangle_{p(x)}$ where $\langle .,.\rangle_{p(x)}$ is the hyperbolic metric. The metric obtained on $M$ is hyperbolic. This is a consequence of the formula definition of sectional curvature which shows that the sectional curvature of the plan $Vect(u,v)$ is the sectional curvature of $Vect(dp_x(u),dp_x(v))$ and a differentiable metric is hyperbolic if is sectional curvature is constant and equal to $-1$.