I am trying to understand how to define a metric for a positively curved two-dimensional space.(adsbygoogle = window.adsbygoogle || []).push({});

I am reading a book and in there it says,

On the surface of a sphere, we can set up a polar coordinate system by picking a pair of antipodal points to be the “north pole” and “south pole” and by picking a geodesic from the north to south pole to be the “prime meridian”. If ##r## is the distance from the north pole, and ##θ## is the azimuthal angle measured relative to the prime meridian, then the distance ds between a point ##(r,θ)## and another nearby point ##(r+dr,θ+dθ)## is given by the relation

$$ds^2 = dr^2 + R^2sin^2(\frac {r}{R})dθ^2$$

where ##R## is the readius of the sphere.

If I think ,flat two dimensional space the metric will be;

$$ds^2 = dr^2 + r^2dθ^2$$

I know the derivation of this so it gives me a bit clue about it,

In this case

So in positively curved space instead of ##rdθ## we will have ##R^2sin^2(\frac {r}{R})dθ## which its kind of obvious.

So anyone who can help to understand the concept maybe an image ?

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# B Spherical Geometry (Two dimension ) Defining a metric

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