- #1

- 14

- 0

*A bowl with axial symmetry is built in flat Euclidean space ##R^3##, and has a radial profile giveb by ##z(r)##, where ##z## is the axis of symmetry and ##r## is the radial distance from the axis. What radial profile ##z(r)## is required for the bowl to have the same intrinsic geometry as the following line element:*

[tex]ds^2=\Bigg(1-\frac{r^2_+}{r^2}\Bigg)^{-1}dr^2+r^2d\phi^2[/tex]

---- MY IDEA -----

The intrinsic geometry of this line element is described by:

[tex]g_{11}=\Bigg(1-\frac{r^2_+}{r^2}\Bigg)^{-1}[/tex] and [tex]g_{22}=r^2[/tex]

But how can I describe the intrinsic geometry of ##z(r)## to equal it to the latter?

Thank you so much for your help!