Geodesic preserving diffeomorphisms of constant curvature spacesConstant curvature manifoldsHyperbolizing geodesic spacesNearly constant curvature implies “nearly isometric” to a space form?Bounding the perimeter of a geodesic triangle in spaces of non-positive curvatureA 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
Constant curvature manifoldsHyperbolizing geodesic spacesNearly constant curvature implies “nearly isometric” to a space form?Bounding the perimeter of a geodesic triangle in spaces of non-positive curvatureA 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 $mathbbR^n$, the 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.
In the first 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 projective-geometry symmetric-spaces geodesics
edited Aug 13 at 7:37
Ben McKay
16k2 gold badges32 silver badges64 bronze badges
asked Aug 10 at 13:16
MKOMKO
7,6043 gold badges35 silver badges73 bronze badges
$endgroup$
add a comment |
$begingroup$
Let $X$ be either Euclidean space $mathbbR^n$, the 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.
In the first 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 projective-geometry symmetric-spaces geodesics
edited Aug 13 at 7:37
Ben McKay
16k2 gold badges32 silver badges64 bronze badges
asked Aug 10 at 13:16
MKOMKO
7,6043 gold badges35 silver badges73 bronze badges
$endgroup$
add a comment |
$begingroup$
Let $X$ be either Euclidean space $mathbbR^n$, the 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.
In the first 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 projective-geometry symmetric-spaces geodesics
edited Aug 13 at 7:37
Ben McKay
16k2 gold badges32 silver badges64 bronze badges
asked Aug 10 at 13:16
MKOMKO
7,6043 gold badges35 silver badges73 bronze badges
$endgroup$
Let $X$ be either Euclidean space $mathbbR^n$, the 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.
In the first 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 projective-geometry symmetric-spaces geodesics
mg.metric-geometry riemannian-geometry projective-geometry symmetric-spaces geodesics
edited Aug 13 at 7:37
Ben McKay
16k2 gold badges32 silver badges64 bronze badges
asked Aug 10 at 13:16
MKOMKO
7,6043 gold badges35 silver badges73 bronze badges
edited Aug 13 at 7:37
Ben McKay
16k2 gold badges32 silver badges64 bronze badges
asked Aug 10 at 13:16
MKOMKO
7,6043 gold badges35 silver badges73 bronze badges
edited Aug 13 at 7:37
Ben McKay
16k2 gold badges32 silver badges64 bronze badges
edited Aug 13 at 7:37
Ben McKay
16k2 gold badges32 silver badges64 bronze badges
edited Aug 13 at 7:37
Ben McKay
16k2 gold badges32 silver badges64 bronze badges
16k2 gold badges32 silver badges64 bronze badges
asked Aug 10 at 13:16
MKOMKO
7,6043 gold badges35 silver badges73 bronze badges
asked Aug 10 at 13:16
MKOMKO
7,6043 gold badges35 silver badges73 bronze badges
asked Aug 10 at 13:16
MKOMKO
7,6043 gold badges35 silver badges73 bronze badges
7,6043 gold badges35 silver badges73 bronze badges
add a comment |
add a comment |
1 Answer
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$begingroup$
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 Aug 10 at 14:45
Ben McKayBen McKay
16k2 gold badges32 silver badges64 bronze badges
$endgroup$
1
$begingroup$
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
Aug 10 at 14:53
add a comment |
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1 Answer
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votes
1 Answer
1
active
oldest
votes
active
oldest
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active
oldest
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 Aug 10 at 14:45
Ben McKayBen McKay
16k2 gold badges32 silver badges64 bronze badges
$endgroup$
1
$begingroup$
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
Aug 10 at 14:53
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 Aug 10 at 14:45
Ben McKayBen McKay
16k2 gold badges32 silver badges64 bronze badges
$endgroup$
1
$begingroup$
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
Aug 10 at 14:53
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 Aug 10 at 14:45
Ben McKayBen McKay
16k2 gold badges32 silver badges64 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 Aug 10 at 14:45
Ben McKayBen McKay
16k2 gold badges32 silver badges64 bronze badges
edited Aug 10 at 14:51
answered Aug 10 at 14:45
Ben McKayBen McKay
16k2 gold badges32 silver badges64 bronze badges
answered Aug 10 at 14:45
Ben McKayBen McKay
16k2 gold badges32 silver badges64 bronze badges
answered Aug 10 at 14:45
Ben McKayBen McKay
16k2 gold badges32 silver badges64 bronze badges
16k2 gold badges32 silver badges64 bronze badges
1
$begingroup$
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
Aug 10 at 14:53
add a comment |
1
$begingroup$
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
Aug 10 at 14:53
1
1
$begingroup$
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
Aug 10 at 14:53
$begingroup$
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
Aug 10 at 14:53
add a comment |
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