General Expression for Round Metric on an N-sphere

[m1rohit]

Homework Statement

I want to know the expression for the round metric of an n-sphere of radius r

Homework Equations

I have obtained this for a 3-sphere
dS^{2}=dr^{2}+r^{2}(d\theta_{1}^{2}+sin^{2}\theta_{1}d\theta_{2}^{2}
+sin^{2}\theta_{1}sin^{2}\theta_{2}d\theta_{3}^{2})

The Attempt at a Solution

what will be the general expression for an n -sphere

 

[Oxvillian]

I have obtained this for a 3-sphere
$$dS^{2}=dr^{2}+r^{2}(d\theta_{1}^{2}+sin^{2}\theta_{1}d\theta_{2}^{2} +sin^{2}\theta_{1}sin^{2}\theta_{2}d\theta_{3}^{2})$$

Looks good to me except that $dr^2$ shouldn't be part of it since the radial direction is not a direction on the n-sphere

Examining the results for n=1, 2 and 3 should reveal the pattern for general n...

 

[bloby]

What is a round metric(in words)? And an induced metric(in words)?
(I don't know the tag for hyperlinks)

http://en.wikipedia.org/wiki/Metric_tensor#The_round_metric_on_a_sphere

http://en.wikipedia.org/wiki/Induced_metric

The benefit of using spherical coordonates(on $\Re^n$)

http://en.wikipedia.org/wiki/N-sphere#Spherical_coordinates

is that the induced metric(on "$S^n$") is just the restriction of the metric for r=cst.
How did you get your answer?
For an arbitrary dimension you can use the coordinate transformations rule for tensors(you know the metric of $\Re^n$ in cartesian coordinate, you know the transformations between them and the spherical coordinates)