Mixing
Projective-invariant differential operator
$begingroup$
This question has been cross-posted to MathOverflow.
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" (are there 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? Also, how can we normalize them so that $S_i[eta]$ is actually invariant under projective transformations of $eta$?
Finally, is there a compact expression for the $n$-dimensional generalization of this derivative?
differential-geometry projective-geometry affine-geometry differential-operators invariance
asked Jan 6 at 22:43
user76284user76284
1,2031126
$endgroup$
add a comment |
$begingroup$
This question has been cross-posted to MathOverflow.
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" (are there 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? Also, how can we normalize them so that $S_i[eta]$ is actually invariant under projective transformations of $eta$?
Finally, is there a compact expression for the $n$-dimensional generalization of this derivative?
differential-geometry projective-geometry affine-geometry differential-operators invariance
asked Jan 6 at 22:43
user76284user76284
1,2031126
$endgroup$
$begingroup$
In your first equations, you state that $gin GLongrightarrow T(gcirc f)=T(f)$. May I ask what this $f$ stands for, an element in $G$ or an arbitrary $mathbb{R}^n$-valued function? Thanks.
$endgroup$
– hypernova
Jan 8 at 23:53
$begingroup$
In that expression, $f$ is an arbitrary function $mathbb{R}^n rightarrow mathbb{R}^n$.
$endgroup$
– user76284
Jan 9 at 0:04
$begingroup$
Thank you for your clarification.
$endgroup$
– hypernova
Jan 9 at 0:09
add a comment |
$begingroup$
This question has been cross-posted to MathOverflow.
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" (are there 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? Also, how can we normalize them so that $S_i[eta]$ is actually invariant under projective transformations of $eta$?
Finally, is there a compact expression for the $n$-dimensional generalization of this derivative?
differential-geometry projective-geometry affine-geometry differential-operators invariance
asked Jan 6 at 22:43
user76284user76284
1,2031126
$endgroup$
This question has been cross-posted to MathOverflow.
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" (are there 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? Also, how can we normalize them so that $S_i[eta]$ is actually invariant under projective transformations of $eta$?
Finally, is there a compact expression for the $n$-dimensional generalization of this derivative?
differential-geometry projective-geometry affine-geometry differential-operators invariance
differential-geometry projective-geometry affine-geometry differential-operators invariance
asked Jan 6 at 22:43
user76284user76284
1,2031126
asked Jan 6 at 22:43
user76284user76284
1,2031126
edited Jan 10 at 20:15
user76284
asked Jan 6 at 22:43
user76284user76284
1,2031126
asked Jan 6 at 22:43
user76284user76284
1,2031126
asked Jan 6 at 22:43
user76284user76284
1,2031126
1,2031126
$begingroup$
In your first equations, you state that $gin GLongrightarrow T(gcirc f)=T(f)$. May I ask what this $f$ stands for, an element in $G$ or an arbitrary $mathbb{R}^n$-valued function? Thanks.
$endgroup$
– hypernova
Jan 8 at 23:53
$begingroup$
In that expression, $f$ is an arbitrary function $mathbb{R}^n rightarrow mathbb{R}^n$.
$endgroup$
– user76284
Jan 9 at 0:04
$begingroup$
Thank you for your clarification.
$endgroup$
– hypernova
Jan 9 at 0:09
add a comment |
$begingroup$
In your first equations, you state that $gin GLongrightarrow T(gcirc f)=T(f)$. May I ask what this $f$ stands for, an element in $G$ or an arbitrary $mathbb{R}^n$-valued function? Thanks.
$endgroup$
– hypernova
Jan 8 at 23:53
$begingroup$
In that expression, $f$ is an arbitrary function $mathbb{R}^n rightarrow mathbb{R}^n$.
$endgroup$
– user76284
Jan 9 at 0:04
$begingroup$
Thank you for your clarification.
$endgroup$
– hypernova
Jan 9 at 0:09
$begingroup$
In your first equations, you state that $gin GLongrightarrow T(gcirc f)=T(f)$. May I ask what this $f$ stands for, an element in $G$ or an arbitrary $mathbb{R}^n$-valued function? Thanks.
$endgroup$
– hypernova
Jan 8 at 23:53
$begingroup$
In your first equations, you state that $gin GLongrightarrow T(gcirc f)=T(f)$. May I ask what this $f$ stands for, an element in $G$ or an arbitrary $mathbb{R}^n$-valued function? Thanks.
$endgroup$
– hypernova
Jan 8 at 23:53
$begingroup$
In that expression, $f$ is an arbitrary function $mathbb{R}^n rightarrow mathbb{R}^n$.
$endgroup$
– user76284
Jan 9 at 0:04
$begingroup$
In that expression, $f$ is an arbitrary function $mathbb{R}^n rightarrow mathbb{R}^n$.
$endgroup$
– user76284
Jan 9 at 0:04
$begingroup$
Thank you for your clarification.
$endgroup$
– hypernova
Jan 9 at 0:09
$begingroup$
Thank you for your clarification.
$endgroup$
– hypernova
Jan 9 at 0:09
add a comment |
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$begingroup$
In your first equations, you state that $gin GLongrightarrow T(gcirc f)=T(f)$. May I ask what this $f$ stands for, an element in $G$ or an arbitrary $mathbb{R}^n$-valued function? Thanks.
$endgroup$
– hypernova
Jan 8 at 23:53
$begingroup$
In that expression, $f$ is an arbitrary function $mathbb{R}^n rightarrow mathbb{R}^n$.
$endgroup$
– user76284
Jan 9 at 0:04
$begingroup$
Thank you for your clarification.
$endgroup$
– hypernova
Jan 9 at 0:09