TITLE: Is there an underlying explanation for the magical powers of the Schwarzian derivative?
QUESTION [129 upvotes]: Given a function $f(z)$ on the complex plane, define the Schwarzian derivative $S(f)$ to be the function
$S(f) = \frac{f'''}{f'} - \frac{3}{2} (\frac{f''}{f'})^2$
Here is a somewhat more conceptual definition, which justifies the terminology.  Define $[f, z, \epsilon]$ to be the cross ratio $[f(z), f(z + \epsilon); f(z + 2\epsilon), f(z + 3\epsilon)]$, and let $[z, \epsilon]$ denote the cross ratio $[z, z + \epsilon, z + 2\epsilon, z + 3\epsilon]$ (in fact this is just 4, but this notation makes my point clearer).  One can ask if $[f, z, \epsilon]$ is well approximated by $[z, \epsilon]$; indeed, it turns out that the error is $o(\epsilon)$.  So one pursues the second order error term and finds that $[f, z, \epsilon] = [z, \epsilon] - 2 S(f)(z) \epsilon^2 + o(\epsilon^2)$.  So the Schwarzian derivative measures the infinitesimal change in cross ratio caused by $f$.  In particular, $S(f)$ is identically zero precisely for Möbius transformations. 
That's all background.  From what I have said so far, the Schwarzian derivative is at best a curiosity.  What is not obvious at first glance is the fact that the Schwarzian derivative has magical powers.  Here are some examples:
First magical power: The Schwarzian derivative is deeply relevant to one dimensional dynamics, stemming from the fact that it behaves in a specific way under compositions.  For example, if $f$ is a smooth function from the unit interval to itself with negative Schwarzian derivative and $n$ critical points, then it has at most n+2 attracting periodic orbits.
Second magical power: It says something profound about the solutions to the Sturm-Liouville equation, $f''(z) + u(z) f(z) = 0$.  If $f_1$ and $f_2$ are two linearly independent solutions, then the ratio $g(z) = f_1(z)/f_2(z)$ satisfies $S(g) = 2u$.
Third magical power: The Schwarzian derivative is the unique projectively invariant 1-cocyle for the diffeomorphism group of $\mathbb{R}P^1$.  This is probably just a restatement of the conceptual definition I gave above, but I'm not sure; in any event, this gives the Schwarzian derivative a great deal of relevance to conformal field theory (or so I'm told).  
I'm sure there are more.  I'm wondering if all of these powers can be explained by some underlying geometric principle.  They all seem vaguely relevant to each other, but the first power in particular seems very hard to relate to the definition in any obvious way.  Does anybody have any insights?

REPLY [5 votes]: Let G(k,n) be the k'th order jet group in n variables which consists of the set of k-jets of local diffeomorphisms of R(n) fixing the origin under the operation of composition. In coordinates, this group operation can be written explicitly using the chain rule.
Now consider G(3,1) and G(2,1) and the obvious projection homomorphism from G(3,1) onto G(2,1) induced by jets. This projection splits, that is, there is an injective homomorphism which imbeds G(2,1) into G(3,1). Identifying the image of this injection with G(2,1), we can form the left (say) coset space G(3,1)/G(2,1). Now the expression for the Schwarzian derivative defines "coordinates" on G(3,1)/G(2,1). The details of these computations can be found on pages 152-153 of the book "An Alternative Approach to Lie Groups and Geometric Structures". This construction generalizes to arbitrary dimensions and is a very special case of defining geometric structures as explained on pages 174-182 of this book. 
To summarize here, these splittings arise from homogeneous spaces and define "geometric connections" which are not necessarily connections in the classical sense (keeping in mind that connections can be defined on general principal and vector bundles and are essentially topological objects). Furthermore, these splittings are built into the definition of a geometric structure and therefore there is no need to search for a "special connection" suitable for some geometric structure. Consequently, the magical powers of the Scwarzian derivative is a special case of the magical powers of "connections". 
As an interesting detail, the obvious left action of G(3,1) on G(3,1)/G(2,1) gives the transformation rule of the Schwarzian derivative found in classical textbooks. In affine case (the action of G(2,n) on G(2,n)/G(1,n)), for instance, we get the well known transformation rule of the "connection components" on the tangent bundle which is historically the starting point of the theory of connections on vector bundles! The action of G(1,n) on G(1,n)/G(0,n) = G(1,n) gives absolute parallelism studied in detail in this book.