Before going further, I would like to mention a convention in writing covariant derivative of tensors in a short form. What I mean is how to understand the following notation where
is a Riemannian metric.
We all know that we can apply covariant derivative to a scalar function, for example, is nothing but the partial derivative with respect to
. However, the notation
is a little bit different. For
, a Riemannian metric, which is also a
tensor, in the full form, we can write
.
The sub indexes in the notation
tell us that we are talking about the
-component of the covariant derivative of
tensor
with respect to the vector field
.
If you have , then you are working on some
-tensor
of the form
.
We now go back to the case . Precisely, one should write
.
Since is a basis for the dual space of a space spanned by the basis
, it is clear to see that the right hand side of the above convention is just the
-component, that is, the coefficient of the term
.
Now we show that whenever
is a Riemannian metric. This is equivalent to show that every coefficients of
equal to zero, in other word,
. Since
is a
-tensor, then we can compute covariant derivative of
as follows
Since is a metric connection, i.e.
one has
.
Thus
.
In other word the -component of
equals to zero. At the last word, in order to calculate covariant derivative of some
-tensors, you need to use the following three properties
.
.
- For
– or
-tensors:
and
.