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

- Exterior Derivative, d
- Covariant Derivative/Connection, ∇
- Lie Derivative, 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.

**Answer**

Short answer:

- 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

**T. J. Willmore,**The definition of Lie derivative**R. Palais,**A definition of the exterior derivative in terms of Lie derivatives**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.

- 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. - 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 - It turns out that their naturality forces them to be
**unique**if we impose on them some basic properties, such as d∘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, [,], ∇, see here.

Also it is interesting that the Bianchi identities for the connection follow from the naturality and the property d∘d=0 for the exterior derivative, see

**Ph. Delanoe,**On Bianchi identities, e.g. here. - 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 🙂

**Attribution***Source : Link , Question Author : Michael Albanese , Answer Author : Community*