Page 356 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 29: Geodesics

Thus, any result proved concerning the usual covariant differentiation, in particular also for the Notes
covariant differentiation along a curve.

 Y  0.
Definition 2. A vector field Y along a curve  is said to be parallel along if   
Note that if Y and Z are parallel along , then g(Y,Z) is constant.

   g(Y,Z)   g    Y,Z   g Y,    Z   0.

Proposition 1. Let :[a,b]  U be a curve into the Riemannian surface (U, g), let u  U, and let
0
Y  Tu U. Then there is a unique vector field Y along  which is parallel along  and satisfies
0
0
Y(a) = Y .
0
Proof. The condition that Y is parallel along  is a pair of linear first-order ordinary differential
equations:
i
y   i jk     j y . j

i
Given initial conditions  i (a)  y , the existence and uniqueness of a solution on [a, b] follows
0
from the theory of ordinary differential equations.
The proposition together with the comment preceding it shows that parallel translation along a
curve  is an isometry between inner-product spaces P : T U  T U.
a
b

Definition 3. A curve  is a geodesic if its tangent   is parallel along :
   0.

 
If  is a geodesic, then   is constant and hence, every geodesic is parametrized proportionally
to arc length. In particular, if     is a reparametrization of , then  is not a geodesic unless
 is a linear map.

Proposition 2. Let (U, g) be a Riemannian surface, let u  U and let 0  Y  T U. Then there is
0
0
u0

and  > 0, and a unique geodesic :( , ) U, such that (0) u ,  0 and (0) Y .   0

Proof. We have:

      
    i i jk  j k  . i
 
Thus, the condition that  is a geodesic can written as a pair of non-linear second order ordinary
differential equations:
j
i
   i jk  (t)     k .


i
i
i

 
Given initial conditions  i (0) u , (0)  y , there is a unique solution on defined on a small
0
0
enough interval (–, ).
Definition 4. Let :[a,b]  U be a curve. We say that  is length minimizing, or L-minimizing,
if:
L  L
 
for all curves  in U such that (a)   (a) and (b)   (b).


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