- #1

cristo

Staff Emeritus

Science Advisor

- 8,122

- 74

A map [itex]\mathbb{R}^3 \times \mathbb{R} \rightarrow \mathbb{R}^3 [/itex] is defined by

[tex] ((x,y,z),t) \longmapsto \left( \frac{x}{z\sinh t+\cosh t},\frac{y}{z\sinh t+\cosh t},\frac{z\cosh t+\sinh t}{z\sinh t+\cosh t}\right) [/tex]

Show that this determines a mapping [itex] f:S^2\times \mathbb{R} \rightarrow S^2 [/itex].

I tried substituiting polar coordinates for S

^{2}in place of (x,y,z) in the above function. Then I figured that if the function's image is a subset of S

^{2}, then the coordinates must satisfy [itex]z=\sqrt{1-x^2-y^2}[/itex]. However, I can't get this to work, and so it's probably not the correct method!

Any help/hints would be greatly appreciated!