Page 358 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 29: Geodesics

Thus, we have: Notes

1   1 i j  1    i j i   1 k i j i y . j

j 
g

 
 

,
ij
ij
ij,k
ij
2  g    s 0  2  g    s 0  2 ij s 0    g    s 0  2 g y    g   




Since Y vanishes at the endpoints, we can substitute into E (0), and integrate by parts the second


term to get:
b d 1

i
i

E (0)       g   g ik,j    k  y . j


 a dt ij 2  

Since:
d i  i g i k i 1 g i k ,


  


dt  g   g   ij,k   g   2  g ij,k  kj,i    
ij
ij
ij
We now see that:
i
m
j

E (0)   b  g   1  g  g  g     k  y dt   b  g    ,Y  dt.


 a    ij 2 mj,k kj,m mk ,j   a   
Since E (0) 0 for all vector fields Y along  which vanish at the endpoints, we conclude that



   0, and  is a geodesic.

 
The Schwartz inequality implies the following inequality between the length and energy
functional for a curve .
Lemma 2. For any curve , we have
2

L  2E (b a),


with equality if and only if  is parametrized proportionally to arc length.
Finally, the last lemma we state to prove Theorem 1, exhibits the relationship between the L and
E functionals.
Lemma 3. A locally energy-minimizing curve is locally length-minimizing. Furthermore, if  is
locally length-minimizing and  is a reparametrization of  by arc length, then  is locally
energy-minimizing.
Proof. Suppose that  is locally energy-minimizing, and let  be a fixed endpoint variation of .
For each s, let  s (t):[a,b] U be a reparametrization of the curve t   (t;s) proportionally to
arc length. Let (t;s)   s (t), then it is not difficult to see, using say the theorem on continuous

dependence on parameters for ordinary differential equations, that  is also smooth. By
2

Lemma 1,  is a geodesic, hence by Lemma 5, L  2E (b a). It follows that:


2
2
2
2

L (0)  L  2E (b a)  2E (0)(b a) 2E (s)(b a)  L (s)  L (s).










Thus,  is locally length-minimizing proving the first statement in the lemma.
Now suppose that  is locally length-minimizing, and let  be a reparametrization of  by arc
length. Then  is also locally length-minimizing, hence for any fixed-endpoint variation  of ,
we have:
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