Page 363 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes Setting s = 0, (14) follows.
Proof of Proposition 5. We compute:

1 b b b
E   2 a    Yg (X,X)dt  a   Yg  Y X,X dt  a   Yg  X Y,X dt
Y

b
  a  g   X Y,X  g  X Y, Y X  dt
Y
b
  a   g   Y Y,X  g    Y , X  Y,X  g  X Y, X Y  dt
X
b d 

 dt,
    g  Y,X  g  Y, X  R(X,Y,Y,X) g   Y, Y 
a dt Y Y X X X 

,
where as above X    and Y = . Now, the first term integrates to  g  Y Y,X  a b  0, and when

we set s = 0, the second term also vanishes since  X X     0. Furthermore, the last term
 
Y  . Hence, we conclude:
becomes  g    Y,  

b 2

E (0)  a      Y  R(X,Y,X,Y) dt. ...(15)

The proposition now follows from (12).
Thus, E (0) can be viewed as a quadratic form in the generator Y . The corresponding symmetric


bilinear form is called the index form of :
b
I(Y,Z)   a   g    Y,   Z  K    g Y,Z  g  ,Y g    ,Z   dt.
It is the Hessian of the functional E, and if E has a local minimum, I is positive semi-definite. We
will also write I(Y) = I(Y, Y).
Definition 6. Let  be a geodesic parametrized by arc length on the Riemannian surface (U, g). A
vector field Y along  is called a Jacobi field, if it satisfies the following differential equation:


    Y K Y   g   ,Y 
  0. 
 
Two points (a) and (b) along a geodesic  are called conjugate along  if there is a non-zero
Jacobi field along  which vanishes at those two points.
The Jacobi field equation is a linear system of second-order differential equations. Hence given
initial data specifying the initial value and initial derivative of Y, a unique solution exists along
the entire geodesic .
Proposition 6. Let  be a geodesic on the Riemannian surface (U, g). Then given two vectors

Z , Z  T  (a) U, there is a unique Jacobi field Y along  such that Y(a) = Z , and    Y(a) Z .
2
2
1
1
In particular, any Jacobi field which is tangent to  is a linear combination of   and t .  The
significance of Jacobi fields is seen in the following two propositions. We say that  is a variation
of  through geodesics if the curves t   (t;s) are geodesics for all s.

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