Away from the four fixed points, negation acts freely and the quotient map is a holomorphic double covering. Near a fixed point, an invariant coordinate is , so the quotient extends across the image as a Riemann surface and the map has ramification index two. The Riemann-Hurwitz formula gives
(1)
hence the quotient has genus zero and is biholomorphic to the Riemann sphere.

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  1. Negation map on a one-dimensional complex torus
  2. Complex torus
  3. Complex manifold
  4. Integrable almost complex structure
  5. Almost complex manifold
  6. Complex structure
  7. Complex geometry
  8. Geometry and topology
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  10. Mathematics
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