# Curvature

**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**

** Area of a sphere = 4 Pi R**^{2} , Area swept out = 4 Pi R^{2}/8

**
**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:

**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 **

**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**