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Hi, All:
Let ## f: X → Y ## be a differentiable map , so that ## Df(x)≠0 ## for all ##x## in ##X##. Then the inverse function
theorem guarantees that every point has a neighborhood where ##f ## restricts to a homeomorphism.
Does anyone know the conditions under which conditions a map like above is a covering map? I'm thinking of the case of the complex exponential ## e^z ## , with ##d/dz(e^z)=e^z ≠0## which is a covering map ## \mathbb C^2 → (\mathbb C-{0} ) ## , but I can't tell if the condition ## df(x)≠ 0 ## is enough to guarantee that ##f ## is a covering map, nor what conditions would make ##f ## into a covering map.
Thanks for any Ideas.
Let ## f: X → Y ## be a differentiable map , so that ## Df(x)≠0 ## for all ##x## in ##X##. Then the inverse function
theorem guarantees that every point has a neighborhood where ##f ## restricts to a homeomorphism.
Does anyone know the conditions under which conditions a map like above is a covering map? I'm thinking of the case of the complex exponential ## e^z ## , with ##d/dz(e^z)=e^z ≠0## which is a covering map ## \mathbb C^2 → (\mathbb C-{0} ) ## , but I can't tell if the condition ## df(x)≠ 0 ## is enough to guarantee that ##f ## is a covering map, nor what conditions would make ##f ## into a covering map.
Thanks for any Ideas.