Several aspects of curvature :

     Consider the 2 - sphere :

Note that the geodesics which go from the noth pole to the south pole cross the
equator at exactly 90°. Consider two such geodesics, side by side, which begin
at the equator and go toward the north pole. Note that the distance between these two
geodesics gets smaller and smaller as the points get closer to the north pole. The point is
that even if these two geodesics start out parallel at the equator, they can not stay
parallel as the points move northward.

The reason why this occurs is that the surface of the sphere is intrinsically curved!

Conclusion: One way to tell whether you are on a curved surface is to construct two initially parallel geodesics and measure their relative separation: if it gets smaller or larger, then you are on an intrinsically curved surface.


     A   quantitative   measure   of   curvature   based   on   what   happens   to   relative   geodesics :  		   Curvature   =   (Change in the angle of approach of two geodesics)/(spatial area over which the change happened)

     Example :    Consider two geodesics which have the property that they leave the equator and arrive at the north pole at 90 Degree difference in    directions .    Then, since they started out at the equator parallel, the change in the directions after the whole trip to the north pole is precisely 90 Degree or \[Pi]/2 radians [recall that there are 2 \[Pi] radians in 360 Degree] .    In addition to this, we need the area swept out between these two north - seeking geodesics :  			 

                                                                                                    2          2
2 Area of a sphere 4 \[Pi] R \[Pi] R
StyleBox[{Area of a sphere = 4 \[Pi] R , Area swept out = ---------------- = ---------- = --------- }, ShowStringCharacters -> True]
8 8 2

Thus, we find that the curvature is given by

     Curvature =    \[Pi]/2/\[Pi] R^2 /2    =    1/( R^2)

This implies that as the sphere gets of larger and larger radius, the curvature of it becomes smaller and smaller. Also, what is very special about the sphere is the constancy of its curvature: the curvature is the same at all points of the spherical surface.

Now for a more general surface, this is not the case as the following example surface shows quite clearly:


     StyleBox[{This surface has variable curvature in both directions .    This is the generic case ... the sphere was special indeed !, Another characterization of curvature which can be shown to be equivalent to the one considered above is the following :}, ShowStringCharacters -> True]

Transport Around a Closed Path

We mention here that the curvature of a surface can be measured by traversing a closed path. If one parallely transports a vector around the path and then compares the transported vector with a copy which stayed at the beginning point, one gets another measure of curvature:

Curvature =

Change in direction of vector during parallel transport
space area circumnavigated