Möbius transformations

Target entity: Möbius transformations
Context triple: [Kleinian group, actsBy, Möbius transformations]

• A. Möbius geometry
  Möbius geometry is a branch of geometry that studies properties of figures invariant under Möbius (conformal) transformations of the extended complex plane or higher-dimensional spheres.
• B. Riemann sphere
  The Riemann sphere is the complex plane plus a point at infinity, forming a one-dimensional complex manifold topologically equivalent to a sphere and used to study meromorphic functions and complex analysis.
• C. Riemann mapping theorem
  The Riemann mapping theorem is a fundamental result in complex analysis stating that any non-empty simply connected open subset of the complex plane (other than the whole plane) can be conformally mapped onto the open unit disk.
• D. Poincaré upper half-plane model
  The Poincaré upper half-plane model is a standard representation of the hyperbolic plane using the complex numbers with positive imaginary part, equipped with a specific metric that makes geodesics appear as semicircles and vertical lines.
• E. Transformations
  "Transformations" is an influential art-critical work by British critic Roger Fry that explores the nature and evolution of modern art and aesthetic experience.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: Möbius transformations
Target entity description: Möbius transformations are conformal automorphisms of the extended complex plane represented by fractional linear functions that map circles and lines to circles and lines.

• A. Möbius geometry
  Möbius geometry is a branch of geometry that studies properties of figures invariant under Möbius (conformal) transformations of the extended complex plane or higher-dimensional spheres.
• B. Riemann sphere
  The Riemann sphere is the complex plane plus a point at infinity, forming a one-dimensional complex manifold topologically equivalent to a sphere and used to study meromorphic functions and complex analysis.
• C. Riemann mapping theorem
  The Riemann mapping theorem is a fundamental result in complex analysis stating that any non-empty simply connected open subset of the complex plane (other than the whole plane) can be conformally mapped onto the open unit disk.
• D. Poincaré upper half-plane model
  The Poincaré upper half-plane model is a standard representation of the hyperbolic plane using the complex numbers with positive imaginary part, equipped with a specific metric that makes geodesics appear as semicircles and vertical lines.
• E. Transformations
  "Transformations" is an influential art-critical work by British critic Roger Fry that explores the nature and evolution of modern art and aesthetic experience.
• F. None of above. chosen