Geodesic preserving diffeomorphisms of constant curvature spacesA question about the proof of Mostow rigidityAre there arbitrarily sparse “lattices” in negatively curved symmetric spaces?Geodesic rays and horofunctionsIsometry groups of Riemannian submersions with totally geodesic fibersNearly constant curvature implies “nearly isometric” to a space form?Bounding the perimeter of a geodesic triangle in spaces of non-positive curvatureDo there exist non-totally geodesic isometric minimal immersions $mathbbH^2rightarrow G/K.$A possible characterization of Euclidean geometry via the curvature of the Median-submanifoldBroken geodesic in Finsler polyhedral spaceA.D. Alexandrov imbedding theorem for metrics with symmetry
Geodesic preserving diffeomorphisms of constant curvature spaces
A question about the proof of Mostow rigidityAre there arbitrarily sparse “lattices” in negatively curved symmetric spaces?Geodesic rays and horofunctionsIsometry groups of Riemannian submersions with totally geodesic fibersNearly constant curvature implies “nearly isometric” to a space form?Bounding the perimeter of a geodesic triangle in spaces of non-positive curvatureDo there exist non-totally geodesic isometric minimal immersions $mathbbH^2rightarrow G/K.$A possible characterization of Euclidean geometry via the curvature of the Median-submanifoldBroken geodesic in Finsler polyhedral spaceA.D. Alexandrov imbedding theorem for metrics with symmetry
$begingroup$
Let $X$ be either Euclidean space $mathbbR^n$, sphere $mathbbS^n$, or hyperbolic space $mathbbH^n$.
I would like to have a classification of all diffeomorphisms $Xto X$ which map every geodesic line to a geodesic line.
Remark. In the firt two cases the group of all such transformations is strictly larger than the group of isometries, but for the hyperbolic space I am not sure.
mg.metric-geometry riemannian-geometry symmetric-spaces geodesics
asked 9 hours ago
MKOMKO
7,5123 gold badges35 silver badges73 bronze badges
$endgroup$
add a comment |
$begingroup$
Let $X$ be either Euclidean space $mathbbR^n$, sphere $mathbbS^n$, or hyperbolic space $mathbbH^n$.
I would like to have a classification of all diffeomorphisms $Xto X$ which map every geodesic line to a geodesic line.
Remark. In the firt two cases the group of all such transformations is strictly larger than the group of isometries, but for the hyperbolic space I am not sure.
mg.metric-geometry riemannian-geometry symmetric-spaces geodesics
asked 9 hours ago
MKOMKO
7,5123 gold badges35 silver badges73 bronze badges
$endgroup$
add a comment |
$begingroup$
Let $X$ be either Euclidean space $mathbbR^n$, sphere $mathbbS^n$, or hyperbolic space $mathbbH^n$.
I would like to have a classification of all diffeomorphisms $Xto X$ which map every geodesic line to a geodesic line.
Remark. In the firt two cases the group of all such transformations is strictly larger than the group of isometries, but for the hyperbolic space I am not sure.
mg.metric-geometry riemannian-geometry symmetric-spaces geodesics
asked 9 hours ago
MKOMKO
7,5123 gold badges35 silver badges73 bronze badges
$endgroup$
Let $X$ be either Euclidean space $mathbbR^n$, sphere $mathbbS^n$, or hyperbolic space $mathbbH^n$.
I would like to have a classification of all diffeomorphisms $Xto X$ which map every geodesic line to a geodesic line.
Remark. In the firt two cases the group of all such transformations is strictly larger than the group of isometries, but for the hyperbolic space I am not sure.
mg.metric-geometry riemannian-geometry symmetric-spaces geodesics
mg.metric-geometry riemannian-geometry symmetric-spaces geodesics
asked 9 hours ago
MKOMKO
7,5123 gold badges35 silver badges73 bronze badges
asked 9 hours ago
MKOMKO
7,5123 gold badges35 silver badges73 bronze badges
asked 9 hours ago
MKOMKO
7,5123 gold badges35 silver badges73 bronze badges
asked 9 hours ago
MKOMKO
7,5123 gold badges35 silver badges73 bronze badges
asked 9 hours ago
MKOMKO
7,5123 gold badges35 silver badges73 bronze badges
7,5123 gold badges35 silver badges73 bronze badges
add a comment |
add a comment |
1 Answer
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For $mathbbR^n$: the fundamental theorem of projective geometry (proof:
https://www3.nd.edu/~andyp/notes/FunThmProjGeom.pdf) says that the bijections of $mathbbR^n$ taking lines to lines are the affine maps $xmapsto Ax+b$ for an invertible matrix $A$ and a constant vector $b$.
For $S^n$: a theorem by the same name shows that the bijections of projective space taking projective lines to projective lines is a projective transformation. One easily uses this to prove the result for the sphere that such a bijection is a linear transformation defined up to positive rescaling, acting on the sphere as the space of rays in a real vector space.
For $mathbbH^n$: Kobayashi defined a Kobayashi pseudometric for projective connections, which determines the usual metric when applied to hyperbolic space, so the unparameterized geodesics determine the metric, and so the diffeomorphisms preserving them are isometries:
S. Kobayashi, Intrinsic distances associated with flat affine or projective structures, J. Fac. Sci. Univ. Tokyo IA 24 (1977), 129--135.
answered 7 hours ago
Ben McKayBen McKay
15.7k2 gold badges31 silver badges63 bronze badges
$endgroup$
$begingroup$
You could also use Hilbert's construction of metrics from cross ratios to prove the result for hyperbolic space, if I remember correctly, and that might get around the need to assume diffeomorphism instead of just bijection.
$endgroup$
– Ben McKay
7 hours ago
add a comment |
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1 Answer
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1 Answer
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active
oldest
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active
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votes
$begingroup$
For $mathbbR^n$: the fundamental theorem of projective geometry (proof:
https://www3.nd.edu/~andyp/notes/FunThmProjGeom.pdf) says that the bijections of $mathbbR^n$ taking lines to lines are the affine maps $xmapsto Ax+b$ for an invertible matrix $A$ and a constant vector $b$.
For $S^n$: a theorem by the same name shows that the bijections of projective space taking projective lines to projective lines is a projective transformation. One easily uses this to prove the result for the sphere that such a bijection is a linear transformation defined up to positive rescaling, acting on the sphere as the space of rays in a real vector space.
For $mathbbH^n$: Kobayashi defined a Kobayashi pseudometric for projective connections, which determines the usual metric when applied to hyperbolic space, so the unparameterized geodesics determine the metric, and so the diffeomorphisms preserving them are isometries:
S. Kobayashi, Intrinsic distances associated with flat affine or projective structures, J. Fac. Sci. Univ. Tokyo IA 24 (1977), 129--135.
answered 7 hours ago
Ben McKayBen McKay
15.7k2 gold badges31 silver badges63 bronze badges
$endgroup$
$begingroup$
You could also use Hilbert's construction of metrics from cross ratios to prove the result for hyperbolic space, if I remember correctly, and that might get around the need to assume diffeomorphism instead of just bijection.
$endgroup$
– Ben McKay
7 hours ago
add a comment |
$begingroup$
For $mathbbR^n$: the fundamental theorem of projective geometry (proof:
https://www3.nd.edu/~andyp/notes/FunThmProjGeom.pdf) says that the bijections of $mathbbR^n$ taking lines to lines are the affine maps $xmapsto Ax+b$ for an invertible matrix $A$ and a constant vector $b$.
For $S^n$: a theorem by the same name shows that the bijections of projective space taking projective lines to projective lines is a projective transformation. One easily uses this to prove the result for the sphere that such a bijection is a linear transformation defined up to positive rescaling, acting on the sphere as the space of rays in a real vector space.
For $mathbbH^n$: Kobayashi defined a Kobayashi pseudometric for projective connections, which determines the usual metric when applied to hyperbolic space, so the unparameterized geodesics determine the metric, and so the diffeomorphisms preserving them are isometries:
S. Kobayashi, Intrinsic distances associated with flat affine or projective structures, J. Fac. Sci. Univ. Tokyo IA 24 (1977), 129--135.
answered 7 hours ago
Ben McKayBen McKay
15.7k2 gold badges31 silver badges63 bronze badges
$endgroup$
$begingroup$
You could also use Hilbert's construction of metrics from cross ratios to prove the result for hyperbolic space, if I remember correctly, and that might get around the need to assume diffeomorphism instead of just bijection.
$endgroup$
– Ben McKay
7 hours ago
add a comment |
$begingroup$
For $mathbbR^n$: the fundamental theorem of projective geometry (proof:
https://www3.nd.edu/~andyp/notes/FunThmProjGeom.pdf) says that the bijections of $mathbbR^n$ taking lines to lines are the affine maps $xmapsto Ax+b$ for an invertible matrix $A$ and a constant vector $b$.
For $S^n$: a theorem by the same name shows that the bijections of projective space taking projective lines to projective lines is a projective transformation. One easily uses this to prove the result for the sphere that such a bijection is a linear transformation defined up to positive rescaling, acting on the sphere as the space of rays in a real vector space.
For $mathbbH^n$: Kobayashi defined a Kobayashi pseudometric for projective connections, which determines the usual metric when applied to hyperbolic space, so the unparameterized geodesics determine the metric, and so the diffeomorphisms preserving them are isometries:
S. Kobayashi, Intrinsic distances associated with flat affine or projective structures, J. Fac. Sci. Univ. Tokyo IA 24 (1977), 129--135.
answered 7 hours ago
Ben McKayBen McKay
15.7k2 gold badges31 silver badges63 bronze badges
$endgroup$
For $mathbbR^n$: the fundamental theorem of projective geometry (proof:
https://www3.nd.edu/~andyp/notes/FunThmProjGeom.pdf) says that the bijections of $mathbbR^n$ taking lines to lines are the affine maps $xmapsto Ax+b$ for an invertible matrix $A$ and a constant vector $b$.
For $S^n$: a theorem by the same name shows that the bijections of projective space taking projective lines to projective lines is a projective transformation. One easily uses this to prove the result for the sphere that such a bijection is a linear transformation defined up to positive rescaling, acting on the sphere as the space of rays in a real vector space.
For $mathbbH^n$: Kobayashi defined a Kobayashi pseudometric for projective connections, which determines the usual metric when applied to hyperbolic space, so the unparameterized geodesics determine the metric, and so the diffeomorphisms preserving them are isometries:
S. Kobayashi, Intrinsic distances associated with flat affine or projective structures, J. Fac. Sci. Univ. Tokyo IA 24 (1977), 129--135.
answered 7 hours ago
Ben McKayBen McKay
15.7k2 gold badges31 silver badges63 bronze badges
edited 7 hours ago
answered 7 hours ago
Ben McKayBen McKay
15.7k2 gold badges31 silver badges63 bronze badges
answered 7 hours ago
Ben McKayBen McKay
15.7k2 gold badges31 silver badges63 bronze badges
answered 7 hours ago
Ben McKayBen McKay
15.7k2 gold badges31 silver badges63 bronze badges
15.7k2 gold badges31 silver badges63 bronze badges
$begingroup$
You could also use Hilbert's construction of metrics from cross ratios to prove the result for hyperbolic space, if I remember correctly, and that might get around the need to assume diffeomorphism instead of just bijection.
$endgroup$
– Ben McKay
7 hours ago
add a comment |
$begingroup$
You could also use Hilbert's construction of metrics from cross ratios to prove the result for hyperbolic space, if I remember correctly, and that might get around the need to assume diffeomorphism instead of just bijection.
$endgroup$
– Ben McKay
7 hours ago
$begingroup$
You could also use Hilbert's construction of metrics from cross ratios to prove the result for hyperbolic space, if I remember correctly, and that might get around the need to assume diffeomorphism instead of just bijection.
$endgroup$
– Ben McKay
7 hours ago
$begingroup$
You could also use Hilbert's construction of metrics from cross ratios to prove the result for hyperbolic space, if I remember correctly, and that might get around the need to assume diffeomorphism instead of just bijection.
$endgroup$
– Ben McKay
7 hours ago
add a comment |
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