Appendix 4B: Hyperbolic Hopf-map

Hopf, Hopf-Abbildung, hyperbolisch, hyperbolische Hopf-Abbildung

The procedure is as follows:

1.

• Hyperbolic parameterization
• Hyperbolic Hopf map
• Rotation such that the initial state lies on z-axis.
• Map in the Euclidean space.

2.

• Transition to the L-state by permutation of two axes (mirroring at the 45° line between the two axes).
• Hyperbolic Hopf map
• Rotation such that the initial state lies on z-axis.
• Map in the Euclidean space.

Calculation

 

Hopf map in different spaces and with mirroring. The hyperbolic Hopf map from the [math]\displaystyle{ \mathbb{H}^2 }[/math] followed by a projection into the [math]\displaystyle{ \mathbb{R}^3 }[/math] yields the same result as the standard Hopf map except for a sign. But in addition to the standard treatment, the Lie-algebra must be rotated such that the projected initial state returns to the z axis after mirroring. Overall, the procedure is considerably longer, which is why in the main article only the treatment in Euclidean space is considered.

 

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