Page 366 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 29: Geodesics

Thus, we conclude that: Notes

b 2
2 
2 
I(Z,Z)   a  f  h Y  dt 0.

0

0

If I(Z,Z) = 0, then f  and hY  0 on [a, b]. Since Y  0 on (a, b], we conclude that h  on (a,
b], and in view of h(b) = f(b) = 0, we get that Z = 0. Thus, I is positive definite.
29.4 Summary
Let (U, g) be a Riemannian surface, and let : I  U be a curve. A vector field along  is a

.
i
2
smooth function Y : I   The covariant derivative of Y  y  along  is the vector field:
i
j
   Y   i i jk y   k  . i
 y  
Note that if Z is any extension of Y, i.e., a any vector field defined on a neighborhood V of
the image   I of  in U, then we have:
 Z    i Z .
  Y     ;i
 Y  0.
 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.

0
 Let :[a,b]  U be a curve into the Riemannian surface (U, g), let u  U, and let Y  Tu 0
0
U. Then there is a unique vector field Y along  which is parallel along  and satisfies
Y(a) = Y .
0
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.

Let (U, g) be a Riemannian surface, let u  U and let 0  Y  T U. Then there is and  > 0,
 0 0 u0


and a unique geodesic :( , ) U, such that (0) u ,  0 and (0) Y .   0
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).

LOVELY PROFESSIONAL UNIVERSITY 359