- THE £-FUNCTION ON EINSTEIN SOLUTIONS AND RICCI SOLITONS 341
Setting O' ~ 2y'T, we haveThose £-geodesics (i.e., the critical points of£) that emanate from the
origin are given by radial paths s = s ( O') which satisfyd^2 s ·
d0' 2 - tanh s = 0,
s (0) =;: 0.Indeed, if Os= v, then[2
8£ ('-y) =Jo V'f ( 2 dO' ds dO' dv - 20' ( v tanh' s dO' ds + tanh s dO' dv)) dO'= fo
2,,fi ( -2v ~:~ + 2v tanh s) dO'.
Multiplying the £-geodesic equation above by j~ and integrating, we getor equivalently,( ~;)2- 2 log cosh s = s' ( 0)^2 ,
ds ~-----
-d (]' = y'a^2 +2logcoshs, ,where a= s' (0). Note that j~ ~ a and for s large, ~~ R:! ffs. We leave it as
an exercise to check that s R:! ;2
= 2r. In comparison, the radial £-geodesicson a cylinder of any dimension sn-l x ~satisfy s = 2ay'T (see (7.23)).
REMARK 7. 7 4. The solutions of the linearized £-geodesic equation ats = 0, i.e.,
d^2 sd0'2 - s = o,
s (0) = 0,
are s ( O') = a sinh O'.
PROBLEM 7.75. Determine the qualitative properties of the reduced dis-
tance R on the Bryant soliton.