Page 357 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes Let :[a,b] U be a curve. A variation of  is a smooth family of curves (t;s):[a,b] (   , ) I


such that (t;0)   (t) for all t [a,b]. For convenience, we will denote derivatives with respect
to t as usual by a dot, and derivatives with respect to s by a prime. The generator of a variation
 is the vector field Y (t) = (t; 0) along . We say that  is a fixed-endpoint variation, if
 (a;s)   (a), and (b;s)   (b) for all s (  , ). Note that the generator of a fixed-endpoint


variation vanishes at the end points. We say that a variation  is normal if its generator Y is
perpendicular to : g  ,Y    0. A curve  is locally L-minimizing if

b
L (s)   a  g     dt
,

has a local minimum at s = 0 for all fixed-endpoint variations . Clearly, an L-minimizing curve
is locally L-minimizing.
If  is locally L-minimizing, then any reparametrization     of  is also locally L-minimizing.

Indeed, if  is any fixed-endpoint variation of , then (t;s)      1 (t);s  is a fixed-endpoint
variation of , and since reparametrization leaves arc length invariant, we see that L (s) = L (s)


which implies that L also has a local minimum at s = 0. Thus, local minimizers of the functional

L are not necessarily parametrized proportionally to arc length. This helps clarify the following
comment: a locally length-minimizing curve is not necessarily a geodesic, but according to the
next theorem that is only because it may not be parametrized proportionally to arc length.
Theorem 1. A locally length-minimizing curve has a geodesic reparametrization.
To prove this theorem, we introduce the energy functional:
1 b
E  a   g    dt
, 
 2
We may now speak of energy-minimizing and locally energy-minimizing curves. Our first
lemma shows the advantage of using the energy rather than the arc length functional: minimizers
of E are parametrized proportionally to arc length.

Lemma 1. A locally energy-minimizing curve is a geodesic.
Proof. Suppose that  is a locally energy-minimizing curve. We first note that if Y is any vector
field along  which vanishes at the endpoints, then setting (t;s)   (t) sY(t), we see that there

is a fixed-endpoint variation of  whose generator is Y. Since  is locally energy-minimizing, we
have:

b 1

E (0)  a  2  g      s 0 dt  0.
,


We now observe that:
d
d
j
j 
j
  s 0     dt     s 0     dt     s 0   y . j
 



i
where Y  y  is the generator of the fixed-endpoint variation , and:
i
   s 0  g ij,k   s 0  g ij,k y .
k 
g
k

ij


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