Metric in Manifold Homework: 3-sphere in 4D Euclidean Space

[kau]

Homework Statement

I am confused if there is any standard way to check what should be the line element $$ds^{2}$$ when the dimensions are more than three ( since we don't have the option to draw things as we usually do in case of 3d or less dimensional cases. I am following Hobson's book. I am giving you an example. consider a 3sphere embedded in 4d euclidean space. show that line element can be written as following

Homework Equations

$ds^{2} = a^2[(d\chi^2)+sin^{2} \chi(d\theta^{2}+sin^{2}\theta d\phi^{2})]$
now here can you introduce the concept of radial distance?? what it would be??
In that case if I have a metric $ds^2=\frac{dr^{2}}{1-(2\mu/r)}+r^{2}[d\theta^2+sin^{2}\theta d\phi^{2}]$
tell me what should be the radial distance between a sphere at r=2$\mu$ and r=3$\mu$??
these are ques from hobson's book basically.
if any of you take the pain to explain me little bit in detail to visualise these stuff i would be grateful.
thanks

The Attempt at a Solution

 

[member 553083]

For the first case involving a 3-sphere embedded in 4D Euclidean space, the line element can be written as follows:$ds^2 = a^2[(d\chi^2)+sin^{2} \chi(d\theta^{2}+sin^{2}\theta d\phi^{2})]$Where a is the radius of the 3-sphere and $\chi, \theta, \phi$ are the usual spherical coordinates. The radial distance between two points on the 3-sphere can then be found by integrating the line element:$r = \int_0^a ds = \int_0^a a^2[(d\chi^2)+sin^{2} \chi(d\theta^{2}+sin^{2}\theta d\phi^{2})]$For the second case involving the metric $ds^2=\frac{dr^{2}}{1-(2\mu/r)}+r^{2}[d\theta^2+sin^{2}\theta d\phi^{2}]$, the radial distance between a sphere at radius $r_1=2\mu$ and $r_2=3\mu$ can be found by integrating the line element from $r_1$ to $r_2$:$r_{12} = \int_{2\mu}^{3\mu} ds = \int_{2\mu}^{3\mu} \frac{dr^{2}}{1-(2\mu/r)}+r^{2}[d\theta^2+sin^{2}\theta d\phi^{2}]$