Isometries between spherical space formsFree actions of finite groups on products of even-dimensional spheresIsometry classification of spherical space formsRealizing a homology by a smooth immersionCombination theorems for discrete subgroups of isometry groupsGeodesic cuffs of pairs of pants in a hyperbolic manifold- why are they disjoint?Groups of equi-quasi-isometric diffeomorphisms of a Riemannian surface of bounded geometryWhich spherical space forms embed in $S^4$?Relation between conjugacy class, quotient isomorphism class, and signature of Fuchsian groups4-manifolds with finite fundamental group and spherical boundaryCan a hyperbolic manifold be a product?
Isometries between spherical space forms
Free actions of finite groups on products of even-dimensional spheresIsometry classification of spherical space formsRealizing a homology by a smooth immersionCombination theorems for discrete subgroups of isometry groupsGeodesic cuffs of pairs of pants in a hyperbolic manifold- why are they disjoint?Groups of equi-quasi-isometric diffeomorphisms of a Riemannian surface of bounded geometryWhich spherical space forms embed in $S^4$?Relation between conjugacy class, quotient isomorphism class, and signature of Fuchsian groups4-manifolds with finite fundamental group and spherical boundaryCan a hyperbolic manifold be a product?
$begingroup$
Let $S^n/Gamma_i,(i=1,2)$ be a $n$-dimensional spherical space form, where $Gamma_i subset SO(n+1)$ is a finite subgroup acting freely on $S^n$.
Suppose $S^n/Gamma_1$ is diffeomorphic to $S^n/Gamma_2$, can we show they are isometric?
gt.geometric-topology
asked yesterday
TotoroTotoro
43327
$endgroup$
add a comment |
$begingroup$
Let $S^n/Gamma_i,(i=1,2)$ be a $n$-dimensional spherical space form, where $Gamma_i subset SO(n+1)$ is a finite subgroup acting freely on $S^n$.
Suppose $S^n/Gamma_1$ is diffeomorphic to $S^n/Gamma_2$, can we show they are isometric?
gt.geometric-topology
asked yesterday
TotoroTotoro
43327
$endgroup$
1
$begingroup$
A big chunk of the details are in Thurston's "Three-Dimensional Geometry and Topology". As Igor mentioned, the primary technicality issue is Lens spaces, and that part isn't as developed in Thurston's book. Bonahon has a lovely proof in that context, and you can also find Bonahon's proof in Hatcher's 3-manifolds notes (on his web page).
$endgroup$
– Ryan Budney
yesterday
$begingroup$
@RyanBudney Thursron's book and Hatcher's notes are about the three dimensional manifolds while the question is for any $n$. I see something missing here.
$endgroup$
– Piotr Hajlasz
yesterday
$begingroup$
De Rham's theorem is for every dimension. For 3d lens spaces there are other proofs, I think.
$endgroup$
– Igor Belegradek
yesterday
$begingroup$
Whoops, I just imagined this was the $n=3$ case. The general case is similar, although the Bonahon proof is particular to $n=3$.
$endgroup$
– Ryan Budney
yesterday
add a comment |
$begingroup$
Let $S^n/Gamma_i,(i=1,2)$ be a $n$-dimensional spherical space form, where $Gamma_i subset SO(n+1)$ is a finite subgroup acting freely on $S^n$.
Suppose $S^n/Gamma_1$ is diffeomorphic to $S^n/Gamma_2$, can we show they are isometric?
gt.geometric-topology
asked yesterday
TotoroTotoro
43327
$endgroup$
Let $S^n/Gamma_i,(i=1,2)$ be a $n$-dimensional spherical space form, where $Gamma_i subset SO(n+1)$ is a finite subgroup acting freely on $S^n$.
Suppose $S^n/Gamma_1$ is diffeomorphic to $S^n/Gamma_2$, can we show they are isometric?
gt.geometric-topology
gt.geometric-topology
asked yesterday
TotoroTotoro
43327
asked yesterday
TotoroTotoro
43327
asked yesterday
TotoroTotoro
43327
asked yesterday
TotoroTotoro
43327
asked yesterday
TotoroTotoro
43327
43327
1
$begingroup$
A big chunk of the details are in Thurston's "Three-Dimensional Geometry and Topology". As Igor mentioned, the primary technicality issue is Lens spaces, and that part isn't as developed in Thurston's book. Bonahon has a lovely proof in that context, and you can also find Bonahon's proof in Hatcher's 3-manifolds notes (on his web page).
$endgroup$
– Ryan Budney
yesterday
$begingroup$
@RyanBudney Thursron's book and Hatcher's notes are about the three dimensional manifolds while the question is for any $n$. I see something missing here.
$endgroup$
– Piotr Hajlasz
yesterday
$begingroup$
De Rham's theorem is for every dimension. For 3d lens spaces there are other proofs, I think.
$endgroup$
– Igor Belegradek
yesterday
$begingroup$
Whoops, I just imagined this was the $n=3$ case. The general case is similar, although the Bonahon proof is particular to $n=3$.
$endgroup$
– Ryan Budney
yesterday
add a comment |
1
$begingroup$
A big chunk of the details are in Thurston's "Three-Dimensional Geometry and Topology". As Igor mentioned, the primary technicality issue is Lens spaces, and that part isn't as developed in Thurston's book. Bonahon has a lovely proof in that context, and you can also find Bonahon's proof in Hatcher's 3-manifolds notes (on his web page).
$endgroup$
– Ryan Budney
yesterday
$begingroup$
@RyanBudney Thursron's book and Hatcher's notes are about the three dimensional manifolds while the question is for any $n$. I see something missing here.
$endgroup$
– Piotr Hajlasz
yesterday
$begingroup$
De Rham's theorem is for every dimension. For 3d lens spaces there are other proofs, I think.
$endgroup$
– Igor Belegradek
yesterday
$begingroup$
Whoops, I just imagined this was the $n=3$ case. The general case is similar, although the Bonahon proof is particular to $n=3$.
$endgroup$
– Ryan Budney
yesterday
1
1
$begingroup$
A big chunk of the details are in Thurston's "Three-Dimensional Geometry and Topology". As Igor mentioned, the primary technicality issue is Lens spaces, and that part isn't as developed in Thurston's book. Bonahon has a lovely proof in that context, and you can also find Bonahon's proof in Hatcher's 3-manifolds notes (on his web page).
$endgroup$
– Ryan Budney
yesterday
$begingroup$
A big chunk of the details are in Thurston's "Three-Dimensional Geometry and Topology". As Igor mentioned, the primary technicality issue is Lens spaces, and that part isn't as developed in Thurston's book. Bonahon has a lovely proof in that context, and you can also find Bonahon's proof in Hatcher's 3-manifolds notes (on his web page).
$endgroup$
– Ryan Budney
yesterday
$begingroup$
@RyanBudney Thursron's book and Hatcher's notes are about the three dimensional manifolds while the question is for any $n$. I see something missing here.
$endgroup$
– Piotr Hajlasz
yesterday
$begingroup$
@RyanBudney Thursron's book and Hatcher's notes are about the three dimensional manifolds while the question is for any $n$. I see something missing here.
$endgroup$
– Piotr Hajlasz
yesterday
$begingroup$
De Rham's theorem is for every dimension. For 3d lens spaces there are other proofs, I think.
$endgroup$
– Igor Belegradek
yesterday
$begingroup$
De Rham's theorem is for every dimension. For 3d lens spaces there are other proofs, I think.
$endgroup$
– Igor Belegradek
yesterday
$begingroup$
Whoops, I just imagined this was the $n=3$ case. The general case is similar, although the Bonahon proof is particular to $n=3$.
$endgroup$
– Ryan Budney
yesterday
$begingroup$
Whoops, I just imagined this was the $n=3$ case. The general case is similar, although the Bonahon proof is particular to $n=3$.
$endgroup$
– Ryan Budney
yesterday
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Yes, diffeomorphic spherical space forms are isometric. This famous result of Georges de Rham can be found in [de Rham, G.
Complexes à automorphismes et homéomorphie différentiable.
Ann. Inst. Fourier Grenoble 2 (1950), 51–67 (1951)]. The main tool is Reidemeister's torsion. For lens spaces the result was proved by W. Franz in 1935.
answered yesterday
Igor BelegradekIgor Belegradek
19.2k143125
$endgroup$
add a comment |
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$begingroup$
Yes, diffeomorphic spherical space forms are isometric. This famous result of Georges de Rham can be found in [de Rham, G.
Complexes à automorphismes et homéomorphie différentiable.
Ann. Inst. Fourier Grenoble 2 (1950), 51–67 (1951)]. The main tool is Reidemeister's torsion. For lens spaces the result was proved by W. Franz in 1935.
answered yesterday
Igor BelegradekIgor Belegradek
19.2k143125
$endgroup$
add a comment |
$begingroup$
Yes, diffeomorphic spherical space forms are isometric. This famous result of Georges de Rham can be found in [de Rham, G.
Complexes à automorphismes et homéomorphie différentiable.
Ann. Inst. Fourier Grenoble 2 (1950), 51–67 (1951)]. The main tool is Reidemeister's torsion. For lens spaces the result was proved by W. Franz in 1935.
answered yesterday
Igor BelegradekIgor Belegradek
19.2k143125
$endgroup$
add a comment |
$begingroup$
Yes, diffeomorphic spherical space forms are isometric. This famous result of Georges de Rham can be found in [de Rham, G.
Complexes à automorphismes et homéomorphie différentiable.
Ann. Inst. Fourier Grenoble 2 (1950), 51–67 (1951)]. The main tool is Reidemeister's torsion. For lens spaces the result was proved by W. Franz in 1935.
answered yesterday
Igor BelegradekIgor Belegradek
19.2k143125
$endgroup$
Yes, diffeomorphic spherical space forms are isometric. This famous result of Georges de Rham can be found in [de Rham, G.
Complexes à automorphismes et homéomorphie différentiable.
Ann. Inst. Fourier Grenoble 2 (1950), 51–67 (1951)]. The main tool is Reidemeister's torsion. For lens spaces the result was proved by W. Franz in 1935.
answered yesterday
Igor BelegradekIgor Belegradek
19.2k143125
edited yesterday
answered yesterday
Igor BelegradekIgor Belegradek
19.2k143125
answered yesterday
Igor BelegradekIgor Belegradek
19.2k143125
answered yesterday
Igor BelegradekIgor Belegradek
19.2k143125
19.2k143125
add a comment |
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$begingroup$
A big chunk of the details are in Thurston's "Three-Dimensional Geometry and Topology". As Igor mentioned, the primary technicality issue is Lens spaces, and that part isn't as developed in Thurston's book. Bonahon has a lovely proof in that context, and you can also find Bonahon's proof in Hatcher's 3-manifolds notes (on his web page).
$endgroup$
– Ryan Budney
yesterday
$begingroup$
@RyanBudney Thursron's book and Hatcher's notes are about the three dimensional manifolds while the question is for any $n$. I see something missing here.
$endgroup$
– Piotr Hajlasz
yesterday
$begingroup$
De Rham's theorem is for every dimension. For 3d lens spaces there are other proofs, I think.
$endgroup$
– Igor Belegradek
yesterday
$begingroup$
Whoops, I just imagined this was the $n=3$ case. The general case is similar, although the Bonahon proof is particular to $n=3$.
$endgroup$
– Ryan Budney
yesterday