Btukfyl

This question was originally asked on Math StackExchange.

Suppose we want a differential operator $T$ acting on functions $mathbb{R}^n rightarrow mathbb{R}^n$ such that

begin{align*}
&T(g) = 0 Longleftrightarrow g in G \
&g in G Longrightarrow T(g circ f) = T(f)
end{align*}

where $G = text{Aff}(n, mathbb{R})$ is the affine group. Consider the operator

$$T(f) = (nabla f)^{-1} cdot nabla nabla f$$

where $nabla f$ is the gradient of $f$ and $nabla nabla f$ is its Hessian. This seems to satisfy the criteria since

$$nabla nabla f = 0 Longleftrightarrow f(x) = A cdot x + b$$

and

begin{align*}
T(A cdot f + b)
&= (nabla (A cdot f + b))^{-1} cdot nabla nabla (A cdot f + b) \
&= (nabla A cdot f)^{-1} cdot nabla nabla A cdot f \
&= (A cdot nabla f)^{-1} cdot nabla A cdot nabla f \
&= (nabla f)^{-1} cdot A^{-1} cdot A cdot nabla nabla f \
&= (nabla f)^{-1} cdot nabla nabla f \
&= T(f)
end{align*}

My question is this: Is there a similar operator that is invariant under the projective group $G = text{PGL}(n, mathbb{R})$? For $G = text{PGL}(1,mathbb{R})$, an example is the Schwarzian derivative

$$S(f) = frac{f'''}{f'} - frac{3}{2} left(frac{f''}{f'}right)^2$$

Projective differential geometry old and new by Ovsienko and Tabachnikov states in chapter 1.3 page 10 that $S(g) = 0$ iff $g$ is a projective transformation and $S(g circ f) = S(f)$ if $g$ is a projective transformation. They also give a multidimensional generalization of the Schwarzian derivative in equation 7.1.6 page 191:

$$L(f)_{ij}^k = sum_ell frac{partial^2 f^ell}{partial x^i partial x^j} frac{partial x^k}{partial f^ell} - frac{1}{n+1} left(delta_j^k frac{partial}{partial x^i} + delta_i^k frac{partial}{partial x^j}right) log J_f$$

where $J_f = det frac{partial f^i}{partial x^j}$ is the Jacobian. However, Schwarps by Pizarro et al. states in section 3.3 page 97 that this "cannot be used to ensure infinitesimally homographic warps as it also vanishes for other functions than homographies" (what are some examples?). Instead, they give a system of 2D Schwarzian equations that "vanish if and only if the warp is a homography" (page 94). These are given in section 4.2 equation 29 page 98:

begin{align*}
S_1[eta] &= eta^x_{uu} eta^y_u - eta^y_{uu} eta^x_u \
S_2[eta] &= eta^x_{vv} eta^y_v - eta^y_{vv} eta^x_v \
S_3[eta] &= (eta^x_{uu} eta^y_v - eta^y_{uu} eta^x_v) + 2(eta^x_{uv} eta^y_u - eta^y_{uv} eta^x_u) \
S_4[eta] &= (eta^x_{vv} eta^y_u - eta^y_{vv} eta^x_u) + 2(eta^x_{uv} eta^y_v - eta^y_{uv} eta^x_v)
end{align*}

What is the geometric intuition behind these equations? Can they be stated more compactly/concisely? How can we normalize them so that $S_i[eta]$ is actually invariant (when nonzero) under projective transformations of $eta$? Finally, is there a compact/closed-form expression for the $n$-dimensional generalization of this derivative?

There's a straightforward abstract answer that you may not like, but, because it clarifies your question and explains a uniform way to answer similar questions, I'll sketch it here.

First, consider a simpler problem of this kind: Suppose that you want to describe the group of isometries of a Riemannian metric $rho$ on a Riemannian $n$-manifold $M$. By definition, a mapping $f:Mto M$ is an isometry if and only if $f^*(rho)-rho =0$, so you could define the operator $T(f) = f^*(rho)-rho$, which takes smooth maps $f$ to sections of $S^2(T^*M)$, and note that $T(f)=0$ if and only if $f$ is an isometry, and, moreover, if $g:Mto M$ is an isometry and $f:Mto M$ is any mapping, we have
$$
T(gcirc f) = (gcirc f)^*(rho)-rho = f^*bigl(g^*(rho)bigr)-rho = f^*(rho)-rho = T(f).
$$
Thus, the differential operator $T$ satisfies the conditions that you want for detecting the group of isometries of $rho$.

Now, consider the slightly more difficult but still manageable case of affine transformations: Let $(M,alpha)$ be a manifold endowed with a (torsion-free) affine connection $alpha$. Now, torsion-free affine connections, unlike Riemannian metrics, are not given by specifying a section of a natural vector bundle over $M$. Instead, there is a natural affine bundle over $M$, call it $mathsf{A}(M)$, that is modeled on the natural vector bundle $TMotimes S^2(T^*M)$, whose sections define the torsion-free affine connections on $M$. The bundle $mathsf{A}(M)$ is natural in the sense that, if $f:Mto M$ is any diffeomorphism, there is an induced canonical bundle isomorphism $mathsf{A}(f):mathsf{A}(M)to mathsf{A}(M)$ such that $mathsf{A}(f)circalpha$ is a section of $mathsf{A}(M)$ that represents the connection $alpha$ pulled back via $f$. We also have $mathsf{A}(gcirc f) = mathsf{A}(f)circ mathsf{A}(g)$, as the canonical map is contravariant. Now, the answer to the problem of characterizing the symmetries of an affine structure $alpha$ on $M$ has a reasonable answer: Simply set
$$
T(f) = mathsf{A}(f)circalpha - alpha,
$$
and this will have all the properties that you want. Note that, because $mathsf{A}(M)$ is modeled on the vector bundle $TMotimes S^2(T^*M)$, the nonlinear differential operator $T$ takes values in the vector bundle $TMotimes S^2(T^*M)$.
When you unravel this for $M=mathbb{R}^n$ and $alpha = alpha_0$, the standard flat affine structure on $mathbb{R}^n$, you get the expression you wrote down above in local coordinates.

Finally, let's come to the case of a manifold of dimension $n>1$ (the case $n=1$ is different) with a (torsion-free) projective structure $(M,pi)$, where, now, $pi$ is a section of a natural affine bundle $mathsf{P}(M)$, that is modeled on the the vector bundle $mathsf{Q}(M)$ that fits into the natural exact sequence
$$
0longrightarrow T^*Mlongrightarrow TMotimes S^2(T^*M)longrightarrow mathsf{Q}(M)longrightarrow 0.
$$
(Note that $mathsf{Q}(M)$ is a vector bundle of rank $tfrac12n(n{-}1)(n{+}2)$. The fact that this rank is $0$ when $n=1$ is why the case $n=1$ is different. Indeed, in dimension $1$ every $2$-jet of a diffeomorphism is the $2$-ject of a projective transformation, so you have to go to $3$-jets to get an equation.)
Again, if $f:Mto M$ is any diffeomorphism, there is a canonically induced bundle mapping $mathsf{P}(f):mathsf{P}(M)to mathsf{P}(M)$, and these bundle maps satisfy $mathsf{P}(gcirc f) = mathsf{P}(f)circ mathsf{P}(g)$.

Now, again, the solution to your problem of characterizing the diffeomorphisms $f:Mto M$ that preserve a given torsion-free projective structure $pi$ is to define
$$
T(f) = mathsf{P}(f)circpi - pi,
$$
and this operator $T$, taking a diffeomorphism $f:Mto M$ to a section of $mathsf{Q}(M)$ (since the difference of two sections of $mathsf{P}(M)$ lies in $mathsf{Q}(M)$), has all the properties that you want. When you write this out in local coordinates, this gives the (second-order) partial differential equations that characterize projective transformations.

Essentially, this approach goes back to Sophus Lie in the 19th century, but it was considerably clarified by the work of Élie Cartan early in the 20th century, in his works on what we would now call Lie pseudo-groups of transformations.