General Expression for Round Metric on an N-sphere

  • Thread starter m1rohit
  • Start date
  • #1
22
0

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
 

Answers and Replies

  • #2
196
22
I have obtained this for a 3-sphere
[tex]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})[/tex]
Looks good to me except that [itex]dr^2[/itex] shouldn't be part of it since the radial direction is not a direction on the n-sphere :smile:

Examining the results for n=1, 2 and 3 should reveal the pattern for general n...
 
  • #3
112
8
What is a round metric(in words)? And an induced metric(in words)?
(I dont 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)
 

Related Threads on General Expression for Round Metric on an N-sphere

Replies
1
Views
822
  • Last Post
Replies
1
Views
849
Replies
4
Views
680
Replies
2
Views
2K
Replies
0
Views
2K
  • Last Post
Replies
8
Views
980
Replies
7
Views
1K
Replies
3
Views
984
Replies
0
Views
9K
Replies
2
Views
484
Top