Global analysis. Exercises 9
1) Find the metric on the sphere S2
 R3
using the following coordinates:
* the coordinates x, y obtained using the stereographic projection;
* the spherical coordinates ,  (for A  S2
,  is the angle between OA and Oz
and  is the angle between OA and Ox, where A is the projection of A to
the plane Oxy).
2) Find the Christoffel symbols for the metric on the sphere S2
 R in the coordinates
of Exercise 1.
3) Let g be the metric on R2
obtained from the metric on the sphere using the
stereographic projection. With respect to the metric g find:
* the middle of the segment AB with A = (0.5; 0.5) and B = (0.9; 0.3);
* the length of the circle (x + 1)2
+ (y - 5)2
= 1.
1