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

.

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