Time to continue the story for covariant derivatives and parallelism, and do what I promised yesterday on tensors.

Fix a smooth manifold with a connection . Then parallel translation along a curve beginning at and ending at leads to an isomorphism , which depends smoothly on . For any , we get isomorphisms depending smoothly on . (Of course, given an isomorphism of vector spaces, there is an isomorphism sending —the important thing is the inverse.)

In the future, I will often omit the superscript because it will cause no confusion.

Thus we have described parallel translation on (potentially mixed) tensors. There are a few things I claim:

- For , we have .
- commutes with contractions.
- commutes with the symmetrization and alternation maps on unmixed (i.e. fully covariant or contravariant) tensors.

The first follows from the description, and so does the second from the parenthetical remark. The third is clear. In particular, the third implies that we can talk about parallel translation of a Riemannian metric or a form.

Now, as in the previous post, it is possible to define a covariant derivative on these tensors. Let and let be a tensor field. Choose a curve tangent to at . Define the **covariant derivative**

For the moment, let’s ignore the question of whether this is well-defined, i.e. that the choice of doesn’t matter. This will follow from the arguments below.

I claim that

- commutes with contractions.

The first is basically the product rule, and follows from the first remark on parallel translation, because can be split into two parts:

and

The second follows from the corresponding of parallel translation. These two together show that the choice of the curve doesn’t matter for any tensor, since it doesn’t matter for zero-tensors (i.e. functions) or vector fields.

Finally, from all this we can also talk about the **covariant derivative of a tensor field along a curve**, using, e.g., the limit. It is the same story though, and I already fear that I have spent too much time on this.

Now we will specialize to the case of a Riemannian metric. Say that a metric and a connection are compatible if

This will be an important notion in the future.

I claim that this is equivalent to saying that **parallel translation preserves inner products on tangent vectors**. Indeed, assume the bolded statement, and fix a curve and parallel vector fields on . Let . Then ( denoting the covariant derivative along a curve)

We can extend locally to vector fields on an open set and so

where is contraction, so

which means that the 2-tensor vanishes on the pair . Since were arbitrary, this completes one half the proof. This argument can be reversed, which gives the other half.

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