Page 350 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 28: Local Intrinsic Geometry of Surfaces

j
j
i
i
Proof. Denoting the coordinates on U by u and the coordinates on U by u , we let    u / u , Notes
j



i
and we find, by the chain rule:

i

j
i


 f    y  f    y ( f)      f  . 



Y  i j i d Y 
z
,
i
i
We define the commutator of two tangent vector fields Y  y  and Z   Equation (5):
i
i
i
i

[Y,Z]   y z i j  z d y j  . j ...(6)
i
Note that
 [Y,Z] f    Z f    Y f. ...(7)
Y
Z
This observation together with Proposition 3.1 are now used to show that the commutator is
reparametrization invariant.



Proposition 2. Let Y and Z be vector fields on U, and let : U   U be a diffeomorphism, then
 

d        d   Y , d Z .  

 Y,Z 



Proof. For any smooth function f on U, we have:
)
)
 f   (f     (f     (f  ) ...(8)




 Y,Z 
d       Y,Z   Y  Z  Z  Y 




      f      f      f    f   f,
    
 
  d Z 
Y  d Z  Z      d Y  d Y    d   Z  d   Y    d Y ,d Z  



and the proposition follows.
We note for future reference that in the proofs of propositions 1 and 2, only the smoothness of

the map  is used, and not the fact that it is a diffeomorphism. The operator Z   Y Z [Y,Z],
also called the Lie derivative, is a differential operator, in the sense that it is linear and satisfies
a Leibniz identity:  Y (fZ)   Y  f Z f  Y Z. However,  Y Z depends on the values of Y in a
neighborhood of a point as can be seen from the fact that it is not linear over functions in Y , but
rather satisfies  fY Z  f Y Z    Z  f Y. Hence the Lie derivative cannot be used as an intrinsic
directional derivative of a vector field Z, which should only depend on the direction vector Y at
a single point .
1
28.3 Covariant Differentiation
Definition 3. Let (U, g) be a Riemannian metric, and let Z be a vector field on U. The covariant
derivative of Z along  is:
i
 i Z   z i j   j ik z k  . j ...(9)
1 Indeed  Y Z as defined does depend only on the value of Y at a single point and satisfies  fY Z  f Y Z.
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