Topology of Curved Space: Understanding Distance on a Positively Curved Sphere

[Niles]

[SOLVED] Topology of curved space

Homework Statement

The distance between a point (r, theta) and a nearby point (r + dr, theta + d\theta) on a positively curved sphere is given by

[tex]
ds^2 = dr^2 + R^2 \sin ^2 (r/R)d\theta ^2
[/tex]

NOTE: I mean that ds^2 = (ds^2). My question is - how do I use this formula? What is what - can you explain it to me?

 

[HallsofIvy]

I'm not sure what your question is. You titled this "Topology of curved space" but topology does not concern itself with distances. You give a formula that involves R but don't say what R is. Apparently your "positively curved sphere" is a sphere of radius R. And if that is the case, then what are your coordinates? In particular, what is "r"?

 

[Niles]

We are dealing with cylindrical coordinates.

So ds is the distance between points (r, theta) and (r+dr, theta + d theta).

Yes, R is the radius of the sphere - I'm sorry I did not mention that earlier.