# Exterior Derivative vs. Covariant Derivative vs. Lie Derivative

In differential geometry, there are several notions of differentiation, namely:

• Exterior Derivative, $d$
• Covariant Derivative/Connection, $\nabla$
• Lie Derivative, $\mathcal{L}$.

I have listed them in order of appearance in my education/in descending order of my understanding of them. Note, there may be others that I am yet to encounter.

Conceptually, I am not sure how these three notions fit together. Looking at their definitions, I can see that there is even some overlap between the collection of objects they can each act on. I am trying to get my head around why there are (at least) three different notions of differentiation. I suppose my confusion can be summarised by the following question.

What does each one do that the other two can’t?

I don’t just mean which objects can they act on that the other two can’t, I would like a deeper explanation (if it exists, which I believe it does). In terms of their geometric intuition/interpretation, does it make sense that we need these different notions?

Note, I have put the reference request tag on this question because I would be interested to find some resources which have a discussion of these notions concurrently, as opposed to being presented as individual concepts.

• the exterior derivative acts on differential forms;
• the Lie derivative acts on any tensors and some other geometric objects (they have to be natural, e.g. a connection, see the paper of P. Petersen below);
• both the exterior and the Lie derivatives don’t require any additional geometric structure: they rely on the differential structure of the manifold;
• the covariant derivative needs a choice of connection which sometimes (e.g. in a presence of a semi-Riemannian metric) can be made canonically;
• there are relationships between these derivatives.

For a longer answer I would suggest the following selection of papers

1. T. J. Willmore, The definition of Lie derivative
2. R. Palais, A definition of the exterior derivative in terms of Lie derivatives
3. P. Petersen, The Ricci and Bianchi Identities

Of course, there is a lot more to say.

Edit. I decided to extend my answer as I believe that there are some essential points which have not been discussed yet.

1. An encyclopedic reference that treats all these derivatives concurrently at a modern level of generality is
I.Kolar, P.W. Michor, J. Slovak, Natural Operations in Differential Geometry (Springer 1993), freely available online here.
I would not even dare to summarize this resource since it has an abysmal deepness and all-round completeness, and indeed covers all the parts of the original question.
Moreover, I believe that the bibliography list of this book contains almost any further relevant reference.
2. As it has been already mentioned by many in this discussion, these operations are intimately related. It cannot be overemphasized that the most important feature that they all share is naturality (they commute with pullback, and this, in particular, makes them coordinate-free).
See KMS cited above and its bibliography, and specifically the following references may be useful:
R. Palais, Natural Operations on Differential Forms, e.g. here or here.
C.L. Terng, Natural Vector Bundles and Natural Differential Operators, e.g. here
3. It turns out that their naturality forces them to be unique if we impose on them some basic properties, such as $d \circ d = 0$ for the exterior derivative. One way to prove that and further references could be found in:
D. Krupka, V. Mikolasova, On the uniqueness of some differential invariants: $d$, $[,]$, $\nabla$, see here.
Also it is interesting that the Bianchi identities for the connection follow from the naturality and the property $d \circ d = 0$ for the exterior derivative, see
Ph. Delanoe, On Bianchi identities, e.g. here.
4. The reference list that I produce here is too far from being complete in any sense. I only would add one classical treatment that I personally used to comprehend some of the fundamental notions related to Lie derivatives (in particular, the Lie derivative of a connection!):
K. Yano, The Theory Of Lie Derivatives And Its Applications, freely available here

Indeed, my comments are speculative and sparse. I wish if this question were answered by someone like P. Michor, to be honest 🙂