Hopf fibration of 3-sphere

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SUMMARY

The Hopf fibration of the 3-sphere consists of fibers that are circles, which do not intersect and collectively form the 3-sphere. The 3-sphere can be represented as the locus of complex pairs (z,w) satisfying |z|² + |w|² = 1, with stereographic projection from the point (0,i) mapping it onto 3D space. The fibers, denoted as S¹, correspond to great circles on the 3-sphere and are characterized by two angles that represent their relative orientation. The discussion clarifies misconceptions about the distances between these circles, emphasizing that the minimum distance between any two circles in the same Hopf fibration varies based on their specific selection.

PREREQUISITES
  • Understanding of Hopf fibration and its mathematical implications
  • Familiarity with stereographic projection and its application in complex analysis
  • Knowledge of the topology of spheres, particularly S² and S³
  • Basic concepts of angles and distances in Euclidean space
NEXT STEPS
  • Explore the mathematical properties of the Hopf fibration in detail
  • Study stereographic projection techniques in complex geometry
  • Investigate the relationship between great circles and their properties in higher-dimensional spheres
  • Examine R. Penrose's concepts of Clifford parallels in "The Road to Reality"
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Mathematicians, physicists, and students interested in topology, complex geometry, and the geometric interpretation of higher-dimensional spaces.

  • #31
cianfa72 said:
Can you explain why they can always be separated in the "ambient" ##\mathbb R^4## ?
Sure. Locally the two circles are one dimensional. When they approach closely enough they look like lines. Suppose one line W is [w, 0, 0, 0] while the other line X is [0,x,0,e]. (w and x are free variables, e is a constant.) If the X line tries to move directly to [0,x,0,-e] then the two lines will intersect at the origin. But if the X line uses the 4th dimension by slipping over to [0,x,e,e] then it can move to [0,x,e,-e] then slip back to [0,x,0,-e]. So it isn't possible to link two circles in R4. It is possible to link two 2-spheres. In N>=3 dimensions one may link (N-2)-spheres.

Great circles are always linked in S3 because they lack the freedom to maneuver like this. If either circle is not a great circle then they cannot be linked. Or at least so I believe.
 
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  • #32
Hornbein said:
Sure. Locally the two circles are one dimensional. When they approach closely enough they look like lines. Suppose one line W is [w, 0, 0, 0] while the other line X is [0,x,0,e]. If the X lines tries to move to [0,x,0,-e] then the two lines will intersect at the origin. (w and x are free variables, e is a constant.) But if the X line uses the 4th dimension by slipping over to [0,x,e,e] then it can move to [0,x,e,-e] then slip back to [0,x,0,-e]. So it isn't possible to link two circles in R4. It is possible to link two 2-spheres. In N dimensions one may link (N-2)-spheres. In two dimensions one may link two 0-spheres.

Great circles are always linked in S3 because they lack the freedom to maneuver like this. If either circle is not a great circle then they cannot be linked. Or at least so I believe.
I think not all great circles. Some of them intersect with each other. The great circles in the Hopf fibration do not intersect with each other, this because they are orbits of the action of SO(2) on S^3. If two orbits intersected then where would the intersection point go under the action of SO(2)? Which orbit would it move on?
 
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  • #33
lavinia said:
I think not all great circles. Some of them intersect with each other. The great circles in the Hopf fibration do not intersect with each other, this because they are orbits of the action of SO(2) on S^3. If two orbits intersected then where would the intersection go under the action of SO(2)?
That's true. If great circles intersect then they are not linked.

Two great circles chosen at random will be linked with probability one. They intersect with probability zero.
 
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  • #34
Hornbein said:
That's true. If great circles intersect then they are not linked.

Two great circles chosen at random will be linked with probability one. They intersect with probability zero.
Can you make that probaility argument more precsise?
 
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