Page 359 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes L 2 L (s)
2


E (0) E  2(b a)  2(b a)  E (s).






Thus,  is locally energy-minimizing.
We note that the same lemma holds if we replace locally energy-minimizing by energy-
minimizing. The proof of Theorem 3 can now be easily completed with the help of Lemmas 1
and 3.
Proof of Theorem 1. Let  be a reparametrization of  by arc length. By Lemma 3,  is locally
energy-minimizing. By Lemma 1,  is a geodesic.
29.2 The Riemann Curvature Tensor

Definition 5. Let X, Y, Z, W be vector fields on a Riemannian surface (U, g). The Riemann
curvature tensor is given by:

R(W, Z, X, Y) g    X , Y Z   [X,Y] Z,W .

We first prove that R is indeed a tensor, i.e., it is linear over functions. Clearly, R is linear in W,
additive in each of the other three variables, and anti-symmetric in X and Y . Thus, it suffices to
prove the following lemma.
Lemma 4. Let X, Y, Z, W be vector fields on a Riemannian surface (U, g). Then we have:
R(W, Z, f X, Y) = R(W, f Z,X, Y) = fR(W, Z, X, Y).
Proof. We have:

  Y Z    fX Z   [fX,Y] Z  f  Y X   Y (f X Z)   f[X,Y] ( Yf)X Z
X
Y
fX




 f  Y Z    Y  f  X Z f  X Z f [X,Y] Z    Y  f  X Z
Y
X
  f   Y Z    X Z   [X,Y] Z  .
Y
X
The first identity follows by taking inner product with W. In order to prove the second identity,
note that:

  Y fZ   X   Yf Z     X f Y  Z   Y  f Z   Y f  X Z  X f  Y Z  f   Y Z.
 

X
X
X
Interchanging X and Y and subtracting we get:
 X , Y  fZ    f Z f[   Y ]Z.
 [X,Y]
X,
On the other hand, we have also:
 [X,Y] fZ   [X,Y]  f Z f  [X,Y] Z.
Thus, we conclude:
 X , Y  fZ   [X,Y] fZ   f [ X , Y ]Z   [X,Y] Z  .

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