Mixing
Surjective differentiable map is an isometry
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This is exercise 1.2 in Svetlana Katok's Fuchsian Groups.
$mathbb{H}$ is the upper half plane (with the hyperbolic metric), and $f:mathbb{H}rightarrowmathbb{H}$ is a surjective $C^1$ map. I want to show $f$ is an isometry (in terms of the hyperbolic metric) if and only if it preserves the Riemannian norm on the tangent bundle of $mathbb{H}$.
One direction I can do (isometry implies norm-preserving), but the other direction is giving me trouble. I've shown that if $f$ is norm-preserving, then it also preserves the length of curves, so that
$$ d(f(z),f(w))le d(z,w) $$
But I can't seem to show there's equality here. In particular, I can't show $f$ is injective. Am I missing something special about the upper half plane?
differential-geometry riemannian-geometry hyperbolic-geometry isometry
asked 11 hours ago
Hempelicious
2010
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up vote
1
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This is exercise 1.2 in Svetlana Katok's Fuchsian Groups.
$mathbb{H}$ is the upper half plane (with the hyperbolic metric), and $f:mathbb{H}rightarrowmathbb{H}$ is a surjective $C^1$ map. I want to show $f$ is an isometry (in terms of the hyperbolic metric) if and only if it preserves the Riemannian norm on the tangent bundle of $mathbb{H}$.
One direction I can do (isometry implies norm-preserving), but the other direction is giving me trouble. I've shown that if $f$ is norm-preserving, then it also preserves the length of curves, so that
$$ d(f(z),f(w))le d(z,w) $$
But I can't seem to show there's equality here. In particular, I can't show $f$ is injective. Am I missing something special about the upper half plane?
differential-geometry riemannian-geometry hyperbolic-geometry isometry
asked 11 hours ago
Hempelicious
2010
Have you used the fact that $f$ must map a geodesic to a geodesic?
– Ted Shifrin
10 hours ago
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
This is exercise 1.2 in Svetlana Katok's Fuchsian Groups.
$mathbb{H}$ is the upper half plane (with the hyperbolic metric), and $f:mathbb{H}rightarrowmathbb{H}$ is a surjective $C^1$ map. I want to show $f$ is an isometry (in terms of the hyperbolic metric) if and only if it preserves the Riemannian norm on the tangent bundle of $mathbb{H}$.
One direction I can do (isometry implies norm-preserving), but the other direction is giving me trouble. I've shown that if $f$ is norm-preserving, then it also preserves the length of curves, so that
$$ d(f(z),f(w))le d(z,w) $$
But I can't seem to show there's equality here. In particular, I can't show $f$ is injective. Am I missing something special about the upper half plane?
differential-geometry riemannian-geometry hyperbolic-geometry isometry
asked 11 hours ago
Hempelicious
2010
This is exercise 1.2 in Svetlana Katok's Fuchsian Groups.
$mathbb{H}$ is the upper half plane (with the hyperbolic metric), and $f:mathbb{H}rightarrowmathbb{H}$ is a surjective $C^1$ map. I want to show $f$ is an isometry (in terms of the hyperbolic metric) if and only if it preserves the Riemannian norm on the tangent bundle of $mathbb{H}$.
One direction I can do (isometry implies norm-preserving), but the other direction is giving me trouble. I've shown that if $f$ is norm-preserving, then it also preserves the length of curves, so that
$$ d(f(z),f(w))le d(z,w) $$
But I can't seem to show there's equality here. In particular, I can't show $f$ is injective. Am I missing something special about the upper half plane?
differential-geometry riemannian-geometry hyperbolic-geometry isometry
differential-geometry riemannian-geometry hyperbolic-geometry isometry
asked 11 hours ago
Hempelicious
2010
asked 11 hours ago
Hempelicious
2010
asked 11 hours ago
Hempelicious
2010
asked 11 hours ago
Hempelicious
2010
asked 11 hours ago
Hempelicious
2010
2010
Have you used the fact that $f$ must map a geodesic to a geodesic?
– Ted Shifrin
10 hours ago
add a comment |
Have you used the fact that $f$ must map a geodesic to a geodesic?
– Ted Shifrin
10 hours ago
Have you used the fact that $f$ must map a geodesic to a geodesic?
– Ted Shifrin
10 hours ago
Have you used the fact that $f$ must map a geodesic to a geodesic?
– Ted Shifrin
10 hours ago
add a comment |
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Have you used the fact that $f$ must map a geodesic to a geodesic?
– Ted Shifrin
10 hours ago