Page 367 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes 29.5 Keywords

Geodesic: A locally energy-minimizing curve is a geodesic.

Schwartz inequality: The Schwartz inequality implies the following inequality between the
length and energy functional for a curve .

29.6 Self Assessment

1. 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 ..................
2. A vector field Y along a curve  is said to be parallel along ..................

3. Let (U, g) be a Riemannian surface, let u  U and let 0  Y  T U. Then there is and  > 0,
0
u0
0
and a unique geodesic .................., such that (0) u ,  0 and (0) Y .   0
4. A locally length-minimizing curve has a ..................
5. The .................. implies the following inequality between the length and energy functional
for a curve .
6. Let  be a .................., and let Y be a Jacobi field. Then, for any vector field Z along , we have

I(Y,Z)   g    Y,Z  a b .

29.7 Review Questions

1. Let g  e g be conformal metrics on U, and let  k ij and  be their Christoffel symbols.
k 
2

ij
Prove that:
k
k 
k
km
k
          g g  m
i
ij
j
j
i
ij
ij
2. Let g  and g be two conformal metrics on U,g  e g, and let K and K be their Gauss

2

curvatures. Prove that:

K  e  2 K   .
Answers: Self Assessment
1.    Y     Z    i Z . 2.  if    Y  0.
;i
3.  :( , ) U, 4. Geodesic reparametrization

5. Schwartz inequality 6. Geodesic

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