- THE RICCI FLOW EQUATION ON 52 AND SOME INTUITION 71
circle of length 27rr and let 51 = 51 (1). We recall the following standard coordinate
systems.
(i) Spherical coordinates. Let 'ljJ E [-~, ~] be the latitude and let e E
[O, 27r) be the longitude on 52 , so that 'l/;(N) = ~ and 'l/;(S) = -~. On 52 -
{N,S} ~ (-~, ~) x 51 the standard metric g 5 2, which has constant curvature 1,
is given by g 52 ( 'l/;, e) = d'l/;^2 + cos^2 'ljJ dB^2 with associated Laplacian 6.. 5 2.
(ii) Cylindrical coordinates. Let (s, B) be the standard coordinates on JR. x
51. The standard cylinder metric is 9cyl = ds^2 +dB^2 with associated Laplacian 6..cyl·
(iii) Euclidean and polar coordinates. Let (x, y) be Euclidean coordinates
on ffi.^2. Let (r, B) be polar coordinates on ffi.^2 - {O}. The Euclidean metric is
9euc = dx^2 + dy^2 = dr^2 + r^2 dB^2 with associated Laplacian 6..euc·
The above metrics are related as follows.
(i') Define the plane to cylinder diffeomorphism T : JR.^2 - {O} -+ JR. x 51 by
T (r, B) = (In r , B) ~ (s(r), B),so that T-^1 (s, e) = (e^5 , e) ~ (r(s), e). We have
dr^2 + r^2 dB^2 = e^25 (ds^2 + dB^2 ).
I n o th er wor d s, ( T -1) 9euc = e 2s 9cyl an d T 9cyl = r - 2 9euc·
(ii') Let a-: 52 - {S}-+ ffi.^2 denote stereographic projection:
(29.1) a-( P )-p-(p,S)S - 1 - (p, S) forpE5^2 -{S},
which is a conformal diffeomorphism with respect to g52 and 9euc· Using (p, S) =
- sin 'l/;(p), we see that
(29.2)
cos'ljJ(p) -1
r(p) ~ la-(p)I = l+sin'l/;(p) = (sec'l/;(p)+tan'l/;(p)).With respect to spherical and polar coordinates, we have that
a-('l/;,B) = ((sec'l/;+tan'l/;)-^1 ,e) ~ (r('l/;),e).Since ~~ = - l_;r 2 and cos 'ljJ = 1 ~~ 2 , we h ave that
4
d'l/;^2 + cos^2 'ljJ dB^2 = (dr^2 + r^2 dB^2 ).(1 + r2)2
That is, (a--^1 )*952 = (1+~2)29euc·
(iii') Mercator projection is the diffeomorphism m: 52 - {N, S}-+ JR. x 51
defined by m =To a-l52_{N,S}; that is,
m('l/;,B) = (-ln(sec'l/;+tan'l/;),B) ~ (s('l/;),e).From this we obtain the relations
cosh s =sec 'ljJ and sinh s = - tan 'l/;.
One computes that ~~ = - sech s, so that
d'l/;^2 + cos^2 'ljJ dB^2 = sech^2 s ( ds^2 + dB^2 ).Equivalently,
(29.3) ( m -1)* g52 = sec h2 s 9cyl an d m * 9cyl = sec 2 "'' 'P g52.