Page 201 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes Applying (8) to (7) we obtain

3
 p X ...........  ...   G X ............  ... ...(9)
n 
...j ...
...n ...

pj 
n   1 
Now we should gather (5), (9), and add the term produced when rq in (2) (or equivalently in (4))
acts upon components of tensor X. As a result we get the following general formula for  p X 1 i ...i r s :
1 j ...j
1 i ...i
s
3
r
3
1 i ...m ...i
 p X 1 i ...i r s   X y 1 j ...j r s    1 G i  pm  X 1 j .... .........j   1  n  X 1 i ....... ...i r s . ...(10)

r

1 j .......... j
1 j ...j
p
s

pj 

1 m  

1 n  
The operator p determined by this formula is called the covariant derivative.
Exercise 1.1: Apply the general formula (10) to a vector field and calculate the covariant derivative
 X .
q
p
Exercise 2.1: Apply the general formula (10) to a covector field and calculate the covariant
derivative  X .
q
p
F . Consider special
Exercise 3.1: Apply the general formula (10) to an operator field and find  p m q
case when  is applied to the Kronecker symbol  q m .
p
Exercise 4.1: Apply the general formula (10) to a bilinear form and find  p a .
qm
Exercise 5.1: Apply the general formula (10) to a tensor product a  x for the case when x is a
vector and a is a covector. Verify formula (a  x) = a  x + a  x.
Exercise 6.1: Apply the general formula (10) to the contraction C(F) for the case when F is an
operator field. Verify the formula C(F) = C(F).
17.9 Concordance of Metric and Connection
Let’s remember that we consider curvilinear coordinates in Euclidean space E. In this space, we
have the scalar product and the metric tensor.
Exercise 1.1: Transform the metric tensor to curvilinear coordinates using transition matrices
and show that here it is given by formula
g = (E , E ). ...(1)
i
ij
j
In Cartesian coordinates all components of the metric tensor are constant since the basis vectors
e , e , e are constant. The covariant derivative (10) in Cartesian coordinates reduces to
2
1
3
differentiation  = /x . Therefore,
p
p
pg = 0. ...(2)
ij
But g is a tensor. If all of its components in some coordinate system are zero, then they are
identically zero in any other coordinate system (explain why). Therefore the identity (2) is valid
in curvilinear coordinates as well.
The identity is known as the concordance condition for the metric g and connection  It is
.
k
ij
ij
very important for general relativity.
Remember that the metric tensor enters into many useful formulas for the gradient, divergency,
rotor, and Laplace operator. What is important is that all of these formulas remain valid in
curvilinear coordinates, with the only difference being that you should understand that  is not
p
the partial derivative /x , but the covariant derivative in the sense of formula (10).
p
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