Understanding S^2 as the Quotient of SO(3) and SO(2)

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SUMMARY

S^2 is established as the quotient space SO(3)/SO(2), a concept rooted in the Hopf fibration. This mathematical relationship can be demonstrated through various methods, including the use of quaternions. Resources such as animations and detailed explanations are available online, providing visual and theoretical insights into this topic. A search for "Hopf fibration" yields extensive information and illustrative materials.

PREREQUISITES
  • Understanding of SO(3) and SO(2) groups
  • Familiarity with the concept of quotient spaces
  • Basic knowledge of the Hopf fibration
  • Introduction to quaternions and their applications in geometry
NEXT STEPS
  • Research the mathematical foundations of the Hopf fibration
  • Explore quaternion algebra and its geometric interpretations
  • Study the properties of SO(3) and SO(2) in relation to topology
  • Examine visual resources and animations that illustrate S^2 as SO(3)/SO(2)
USEFUL FOR

Mathematicians, physicists, and students studying topology and geometry, particularly those interested in the applications of the Hopf fibration and the relationships between different special orthogonal groups.

Lonewolf
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I've seen [itex]S^2[/itex] written as the quotient [itex]SO(3)/SO(2)[/itex]. Can someone run me through how to show this, or point to somewhere that does, as I've only seen it stated?
 
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It's called the Hopf fibration. There are many ways to realize it, one of which uses quarternions I recall. Anyway, a google for hopf fibration will give you more than I can remember off the top of my head, and there's even an animation illustrating it somewhere.
 
Thanks, it helped having something to search for!
 

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