- THE .C-LENGTH AND DISTANCE FOR A STATIC METRIC 287
solution to the heat equation, which is a delta function at 7 = 0 at a point
p EM, we take 71 = 0 so as to define
(7.2) £ ('"'!) ~ } {T Id^12
0VT d;
9d7for C^1 -paths 'Y: [O, f] --+ M from p to arbitrary points q EM.
From (7.1) we see that the critical points 'Y are of the form"( (7) = /3 (2v'i),
where /3 is a constant speed geodesic. Hence, in Euclidean space, 'Y ( 7) =
2ftV for some V E Rn, and its graph ( 'Y ( 7) , 7) is a parabola. This is one
justification for the ft factor in (7.2): parabolas are more suited to the
heat equation. Note that
0 d"( d/3 A
(7.3) r--+O hm vr-d 7 (7) = -d O" (0) E TpM.EXERCISE 7.1. Show that if 'Y is a critical point of (7.2), then
1
\JxX +-X = 0
27where X -- dry dr"
SOLUTION. From (7.1) we find that the critical points satisfy\J ,;TX ( vrX) = 0.
Given a basepoint p (at time 0) define a space-time distance function
on M x (O,oo) by
L (q, f) ~inf£('"'!),
'Y
where the infimum is taken over all C^1 -paths 'Y : [O, f] --+ M with 'Y (0) = p
and 'Y (f) = q. An elementary computation using (7.1) yields
(7.4) L ( f) = d (p, q)
2
q, 2y'r 'where d is the distance with respect to g and the infimum of£ ( "() is obtained
by a minimal geodesic 'Y from p to q with
(7.5)1 d (p, q)
ft 2y'r.With the Euclidean heat kernel in mind we define the reduced distance:
(7.6) e ( q, f) =. L 2y'r (q, f) = d (p, 4f q)2
If we wish to remove the time-dependence in (7.6), then we may define the
enlarged distance: