Page 177 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes

Task Prove the consistence of formula (1).
Interpret this formula as the contraction of the tensor product ax.

15.10 Raising and Lowering Indices

Suppose that X is some tensor of type (r, s). Let’s choose its -th lower index: X ......... . The symbols
...k...
used for the other indices are of no importance. Therefore, we denoted them by dots. Then let’s
consider the tensor product Y = g  X:

Y ...k....  g X .......... . ...(1)
...pq...
pq
....k...
Here g is the dual metric tensor with the components g . In the next step, let’s contract (1) with
pq
respect to the pair of indices k and q. For this purpose we replace them both by s and perform the
summation:
3
........ 
X ...p...  g X ........ . ...(2)
ps
...s...
s 1

This operation (2) is called the index raising procedure. It is invertible. The inverse operation is
called the index lowering procedure:
3
...p... 
X .........  g X ....s.... . ...(3)
ps
..........
s 1

Like (2), the index lowering procedure (3) comprises two tensorial operations: the tensor product
and contraction.
15.11 Some Special Tensors and some useful Formulas
Kronecker symbol is a well known object. This is a two-dimensional array representing the unit
matrix. It is determined as follows:


i
   1 0 for i  j, ...(1)
j
j
 for i 
We can determine two other versions of Kronecker symbol:
 1 for i  j,
  d   ...(2)
ij
ij
 0 for i  j
Exercise 15.25: Prove that definition (1) is invariant under a change of basis, if we interpret the
Kronecker symbol as a tensor. Show that both definitions in (2) are not invariant under a change
of basis.

Exercise 15.26: Lower index i of tensor (1). What tensorial object do you get as a result of this
operation?

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