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 $\mathbb {H} ^{2}$ followed by a projection into the $\mathbb {R} ^{3}$ 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.

 

<- Chapter 4

back to top

---

---

This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.