12) The Riemann Sphere and Mobius Transformations Flashcards
How are multiplication and translation by constants interpreted geometrically in ℂ
How is the map f(z)=1/z extended to ℂ∞
What is the Mobius Transformation
What is the Mobius Function at infinity
What is the group of Mobius Transformations
(Binary operation given by composition)
Describe the proof that the group of Mobius Transformations is a group
What is the nature of the map Φ:GL(2,ℂ)→M
It is a surjective homomorphism
What is the kernel of the map Φ and how is M related to GL(2,ℂ)
What is the projective general linear group
What is the projective special linear group
How unique is the matrix A representing a Möbius transformation f
How can every Möbius transformation f∈M be expressed
What is a circle or line in ℂ∞
What is the image of a circle or line under a Möbius transformation f∈M
Do Möbius transformations always preserve the center of a circle
How does the group M act on the set of circles and lines in ℂ∞
How can we map one triple of points to another using a Möbius transformation
Describe the proof of how we map one triple of points to another using a Möbius transformation
How are any three distinct points in ℂ∞
related to circles and lines
What can be concluded about a Möbius transformation f∈M that fixes three distinct point
Is there a unique Möbius transformation that maps one triple of points to another
How many fixed points does a non-identity Möbius transformation have and what are they conjugate to
What is the classification of Mobius transformations
How is a non-identity Möbius transformation f∈M classified based on its corresponding matrix A∈SL(2,C)
How is a non-identity Möbius transformation f∈M classified based on the trace of its corresponding matrix A∈SL(2,C)
What is the cross ratio of 4 distinct points
How do you calculate the cross ratio
Does multiplying the distinct points affect the cross ratio
When do distinct points lie on a circle or line
