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What is the classification of (3, 4)-pseudoriemannian manifolds?
0
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Have mathematicians found a comprehensive list of all pseudoriemannian manifolds with 3 spacelike and 4 timelike dimensions up to diffeomorphism?
differential-geometry differential-topology
asked Jan 31 at 19:42
ErotemeObelusErotemeObelus
685721
$endgroup$
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show 2 more comments
0
$begingroup$
Have mathematicians found a comprehensive list of all pseudoriemannian manifolds with 3 spacelike and 4 timelike dimensions up to diffeomorphism?
differential-geometry differential-topology
asked Jan 31 at 19:42
ErotemeObelusErotemeObelus
685721
$endgroup$
2
$begingroup$
I don't think such a classification exists, but there are many. For example, the product of any four-manifold with any three-manifold. There are also non-products, such as $S^7$.
$endgroup$
– Michael Albanese
Jan 31 at 22:45
$begingroup$
@MichaelAlbanese does $S^7$ have four timelike dimensions and three spacelike dimensions?
$endgroup$
– ErotemeObelus
Feb 1 at 2:11
$begingroup$
@TomislavOstojich: The point is that $S^7$ is a parallelizable manifold (its tangent bundle is trivial). Every parallelizable $n$-dimensional manifold admits a pseudo-Riemannian metric of any signature $(p,q)$ with $p+q=n$. One can also prove that for every finitely presented group $G$ there exists a compact pseudo-Riemannian $(3,4)$-manifold with fundamental group isomorphic to $G$. Hence, a meaningful classification is impossible.
$endgroup$
– Moishe Kohan
Feb 2 at 12:51
$begingroup$
@MoisheCohen is $S^7$ with a pseudoriemannian metric an exotic sphere?
$endgroup$
– ErotemeObelus
Feb 2 at 14:20
$begingroup$
@TomislavOstojich: If you want so, you can use an exotic sphere. But you can also take it to be the standard sphere. Both are parallelizable, this is all what you need.
$endgroup$
– Moishe Kohan
Feb 2 at 14:33
|
show 2 more comments
0
0
0
$begingroup$
Have mathematicians found a comprehensive list of all pseudoriemannian manifolds with 3 spacelike and 4 timelike dimensions up to diffeomorphism?
differential-geometry differential-topology
asked Jan 31 at 19:42
ErotemeObelusErotemeObelus
685721
$endgroup$
Have mathematicians found a comprehensive list of all pseudoriemannian manifolds with 3 spacelike and 4 timelike dimensions up to diffeomorphism?
differential-geometry differential-topology
differential-geometry differential-topology
asked Jan 31 at 19:42
ErotemeObelusErotemeObelus
685721
asked Jan 31 at 19:42
ErotemeObelusErotemeObelus
685721
asked Jan 31 at 19:42
ErotemeObelusErotemeObelus
685721
asked Jan 31 at 19:42
ErotemeObelusErotemeObelus
685721
asked Jan 31 at 19:42
ErotemeObelusErotemeObelus
685721
685721
2
$begingroup$
I don't think such a classification exists, but there are many. For example, the product of any four-manifold with any three-manifold. There are also non-products, such as $S^7$.
$endgroup$
– Michael Albanese
Jan 31 at 22:45
$begingroup$
@MichaelAlbanese does $S^7$ have four timelike dimensions and three spacelike dimensions?
$endgroup$
– ErotemeObelus
Feb 1 at 2:11
$begingroup$
@TomislavOstojich: The point is that $S^7$ is a parallelizable manifold (its tangent bundle is trivial). Every parallelizable $n$-dimensional manifold admits a pseudo-Riemannian metric of any signature $(p,q)$ with $p+q=n$. One can also prove that for every finitely presented group $G$ there exists a compact pseudo-Riemannian $(3,4)$-manifold with fundamental group isomorphic to $G$. Hence, a meaningful classification is impossible.
$endgroup$
– Moishe Kohan
Feb 2 at 12:51
$begingroup$
@MoisheCohen is $S^7$ with a pseudoriemannian metric an exotic sphere?
$endgroup$
– ErotemeObelus
Feb 2 at 14:20
$begingroup$
@TomislavOstojich: If you want so, you can use an exotic sphere. But you can also take it to be the standard sphere. Both are parallelizable, this is all what you need.
$endgroup$
– Moishe Kohan
Feb 2 at 14:33
|
show 2 more comments
2
$begingroup$
I don't think such a classification exists, but there are many. For example, the product of any four-manifold with any three-manifold. There are also non-products, such as $S^7$.
$endgroup$
– Michael Albanese
Jan 31 at 22:45
$begingroup$
@MichaelAlbanese does $S^7$ have four timelike dimensions and three spacelike dimensions?
$endgroup$
– ErotemeObelus
Feb 1 at 2:11
$begingroup$
@TomislavOstojich: The point is that $S^7$ is a parallelizable manifold (its tangent bundle is trivial). Every parallelizable $n$-dimensional manifold admits a pseudo-Riemannian metric of any signature $(p,q)$ with $p+q=n$. One can also prove that for every finitely presented group $G$ there exists a compact pseudo-Riemannian $(3,4)$-manifold with fundamental group isomorphic to $G$. Hence, a meaningful classification is impossible.
$endgroup$
– Moishe Kohan
Feb 2 at 12:51
$begingroup$
@MoisheCohen is $S^7$ with a pseudoriemannian metric an exotic sphere?
$endgroup$
– ErotemeObelus
Feb 2 at 14:20
$begingroup$
@TomislavOstojich: If you want so, you can use an exotic sphere. But you can also take it to be the standard sphere. Both are parallelizable, this is all what you need.
$endgroup$
– Moishe Kohan
Feb 2 at 14:33
2
2
$begingroup$
I don't think such a classification exists, but there are many. For example, the product of any four-manifold with any three-manifold. There are also non-products, such as $S^7$.
$endgroup$
– Michael Albanese
Jan 31 at 22:45
$begingroup$
I don't think such a classification exists, but there are many. For example, the product of any four-manifold with any three-manifold. There are also non-products, such as $S^7$.
$endgroup$
– Michael Albanese
Jan 31 at 22:45
$begingroup$
@MichaelAlbanese does $S^7$ have four timelike dimensions and three spacelike dimensions?
$endgroup$
– ErotemeObelus
Feb 1 at 2:11
$begingroup$
@MichaelAlbanese does $S^7$ have four timelike dimensions and three spacelike dimensions?
$endgroup$
– ErotemeObelus
Feb 1 at 2:11
$begingroup$
@TomislavOstojich: The point is that $S^7$ is a parallelizable manifold (its tangent bundle is trivial). Every parallelizable $n$-dimensional manifold admits a pseudo-Riemannian metric of any signature $(p,q)$ with $p+q=n$. One can also prove that for every finitely presented group $G$ there exists a compact pseudo-Riemannian $(3,4)$-manifold with fundamental group isomorphic to $G$. Hence, a meaningful classification is impossible.
$endgroup$
– Moishe Kohan
Feb 2 at 12:51
$begingroup$
@TomislavOstojich: The point is that $S^7$ is a parallelizable manifold (its tangent bundle is trivial). Every parallelizable $n$-dimensional manifold admits a pseudo-Riemannian metric of any signature $(p,q)$ with $p+q=n$. One can also prove that for every finitely presented group $G$ there exists a compact pseudo-Riemannian $(3,4)$-manifold with fundamental group isomorphic to $G$. Hence, a meaningful classification is impossible.
$endgroup$
– Moishe Kohan
Feb 2 at 12:51
$begingroup$
@MoisheCohen is $S^7$ with a pseudoriemannian metric an exotic sphere?
$endgroup$
– ErotemeObelus
Feb 2 at 14:20
$begingroup$
@MoisheCohen is $S^7$ with a pseudoriemannian metric an exotic sphere?
$endgroup$
– ErotemeObelus
Feb 2 at 14:20
$begingroup$
@TomislavOstojich: If you want so, you can use an exotic sphere. But you can also take it to be the standard sphere. Both are parallelizable, this is all what you need.
$endgroup$
– Moishe Kohan
Feb 2 at 14:33
$begingroup$
@TomislavOstojich: If you want so, you can use an exotic sphere. But you can also take it to be the standard sphere. Both are parallelizable, this is all what you need.
$endgroup$
– Moishe Kohan
Feb 2 at 14:33
|
show 2 more comments
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2
$begingroup$
I don't think such a classification exists, but there are many. For example, the product of any four-manifold with any three-manifold. There are also non-products, such as $S^7$.
$endgroup$
– Michael Albanese
Jan 31 at 22:45
$begingroup$
@MichaelAlbanese does $S^7$ have four timelike dimensions and three spacelike dimensions?
$endgroup$
– ErotemeObelus
Feb 1 at 2:11
$begingroup$
@TomislavOstojich: The point is that $S^7$ is a parallelizable manifold (its tangent bundle is trivial). Every parallelizable $n$-dimensional manifold admits a pseudo-Riemannian metric of any signature $(p,q)$ with $p+q=n$. One can also prove that for every finitely presented group $G$ there exists a compact pseudo-Riemannian $(3,4)$-manifold with fundamental group isomorphic to $G$. Hence, a meaningful classification is impossible.
$endgroup$
– Moishe Kohan
Feb 2 at 12:51
$begingroup$
@MoisheCohen is $S^7$ with a pseudoriemannian metric an exotic sphere?
$endgroup$
– ErotemeObelus
Feb 2 at 14:20
$begingroup$
@TomislavOstojich: If you want so, you can use an exotic sphere. But you can also take it to be the standard sphere. Both are parallelizable, this is all what you need.
$endgroup$
– Moishe Kohan
Feb 2 at 14:33