Page 362 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 29: Geodesics

Corollary 1. The Riemann curvature tensor of any metric g on a surface is given by: Notes

R ijkl  K  g g  g g jk  . ...(11)
ik
il
i l
Proof. Denote the right-hand side of (11) by S , and note that it satisfies (1). Thus, the same
ijkl
comment which follows Proposition 8 applies and the only non-zero components of S are
ijkl
determined by S 1212 :
S 1212 = –S 2112 = S 2121 = –S 1221 .
In view of (11), we have R 1212 = S 1212 , thus it follows that R = S ijkl
ijkl
In particular, we conclude that:

R(Z,W,X,Y) K g(W,X)g(Z,Y) g(W,Y)g(Z,X) .  ...(12)

29.3 The Second Variation of Arc length

In this section, we study the additional condition E (0) 0 necessary for a minimum. This leads



to the notion of Jacobi fields and conjugate points.
Proposition 5. Let  :[a,b] U be a geodesic parametrized by arc length on the Riemannian
surface (U, g), and let  be a fixed-endpoint variation of  with generator Y. Then, we have:

b 2 2 2

E (0)     Y  K    Y   g   ,Y   dt, ...(13)
 a  
where K is the Gauss curvature of g.
Before we prove this proposition, we offer a second proof of the first variation formula:


E (0)   a b  g    ,Y  dt, ...(14)
 

which is more in spirit with our derivation of the second variation formula. First note that if 

is a fixed-endpoint variation of  with generator  = Y, and with   X, then [X, Y] = 0. Here Y
denotes the vector field  along  rather than just along . Indeed, since X = d (d/dt) and
Y = d(d/ds), it follows, that for any smooth function f on U, we have

 d d 
,
 [X,Y] f    dt ds  f   0.

,
i
In view of the symmetry    i kj this implies:
jk

 Y X   X Y [X,Y] 0.

We can now calculate:
1 b b
E (s)  2  Y  g(X,X)dt  a   g  Y X,X dt  a   g  X ,X dt

Y

b d b b
b
a 
  a dt g(Y,X)dt   a  g Y, X X dt  g(Y,X)  a  g Y, X X dt

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