Does ${rm Iso}(Mtimes N)={rm Iso}(M)times {rm Iso}(N)$ hold for product metrics?
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I know that ${rm Iso} (Mtimes N)={rm Iso}(M)times {rm Iso}(N)$ is not generally correct where $M$ and $N$ are smooth Riemannian manifolds; but
Does ${rm Iso} (Mtimes N)={rm Iso}(M)times {rm Iso}(N)$ hold for product metrics?
what about warp-product metrics?
differential-geometry riemannian-geometry
asked Dec 6 '18 at 7:47
C.F.GC.F.G
1,4301821
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add a comment |
$begingroup$
I know that ${rm Iso} (Mtimes N)={rm Iso}(M)times {rm Iso}(N)$ is not generally correct where $M$ and $N$ are smooth Riemannian manifolds; but
Does ${rm Iso} (Mtimes N)={rm Iso}(M)times {rm Iso}(N)$ hold for product metrics?
what about warp-product metrics?
differential-geometry riemannian-geometry
asked Dec 6 '18 at 7:47
C.F.GC.F.G
1,4301821
$endgroup$
$begingroup$
What do you mean by "for product metrics"? $M$ and $N$ should both be non-trivial Riemannian products of non-flat factors?
$endgroup$
– Anthony Carapetis
Dec 6 '18 at 9:42
1
$begingroup$
This seems relevant.
$endgroup$
– Anthony Carapetis
Dec 6 '18 at 9:48
add a comment |
$begingroup$
I know that ${rm Iso} (Mtimes N)={rm Iso}(M)times {rm Iso}(N)$ is not generally correct where $M$ and $N$ are smooth Riemannian manifolds; but
Does ${rm Iso} (Mtimes N)={rm Iso}(M)times {rm Iso}(N)$ hold for product metrics?
what about warp-product metrics?
differential-geometry riemannian-geometry
asked Dec 6 '18 at 7:47
C.F.GC.F.G
1,4301821
$endgroup$
I know that ${rm Iso} (Mtimes N)={rm Iso}(M)times {rm Iso}(N)$ is not generally correct where $M$ and $N$ are smooth Riemannian manifolds; but
Does ${rm Iso} (Mtimes N)={rm Iso}(M)times {rm Iso}(N)$ hold for product metrics?
what about warp-product metrics?
differential-geometry riemannian-geometry
differential-geometry riemannian-geometry
asked Dec 6 '18 at 7:47
C.F.GC.F.G
1,4301821
asked Dec 6 '18 at 7:47
C.F.GC.F.G
1,4301821
asked Dec 6 '18 at 7:47
C.F.GC.F.G
1,4301821
asked Dec 6 '18 at 7:47
C.F.GC.F.G
1,4301821
asked Dec 6 '18 at 7:47
C.F.GC.F.G
1,4301821
1,4301821
$begingroup$
What do you mean by "for product metrics"? $M$ and $N$ should both be non-trivial Riemannian products of non-flat factors?
$endgroup$
– Anthony Carapetis
Dec 6 '18 at 9:42
1
$begingroup$
This seems relevant.
$endgroup$
– Anthony Carapetis
Dec 6 '18 at 9:48
add a comment |
$begingroup$
What do you mean by "for product metrics"? $M$ and $N$ should both be non-trivial Riemannian products of non-flat factors?
$endgroup$
– Anthony Carapetis
Dec 6 '18 at 9:42
1
$begingroup$
This seems relevant.
$endgroup$
– Anthony Carapetis
Dec 6 '18 at 9:48
$begingroup$
What do you mean by "for product metrics"? $M$ and $N$ should both be non-trivial Riemannian products of non-flat factors?
$endgroup$
– Anthony Carapetis
Dec 6 '18 at 9:42
$begingroup$
What do you mean by "for product metrics"? $M$ and $N$ should both be non-trivial Riemannian products of non-flat factors?
$endgroup$
– Anthony Carapetis
Dec 6 '18 at 9:42
1
1
$begingroup$
This seems relevant.
$endgroup$
– Anthony Carapetis
Dec 6 '18 at 9:48
$begingroup$
This seems relevant.
$endgroup$
– Anthony Carapetis
Dec 6 '18 at 9:48
add a comment |
1 Answer
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oldest
votes
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Example: $M=N=mathbb{R}$ with the standard metric. Then $operatorname{Isom}mathbb{E}^2$ is a 3-dimensional Lie group, so obviously cannot be the Cartesian square of $operatorname{Isom}mathbb{E}^1$.
answered Dec 6 '18 at 8:44
user10354138user10354138
7,4322925
$endgroup$
$begingroup$
Thank you for your answer. What can be say in compact case?
$endgroup$
– C.F.G
Dec 7 '18 at 5:00
1
$begingroup$
For example, see the link given by @AnthonyCarapetis
$endgroup$
– user10354138
Dec 7 '18 at 15:25
add a comment |
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1 Answer
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1 Answer
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$begingroup$
Example: $M=N=mathbb{R}$ with the standard metric. Then $operatorname{Isom}mathbb{E}^2$ is a 3-dimensional Lie group, so obviously cannot be the Cartesian square of $operatorname{Isom}mathbb{E}^1$.
answered Dec 6 '18 at 8:44
user10354138user10354138
7,4322925
$endgroup$
$begingroup$
Thank you for your answer. What can be say in compact case?
$endgroup$
– C.F.G
Dec 7 '18 at 5:00
1
$begingroup$
For example, see the link given by @AnthonyCarapetis
$endgroup$
– user10354138
Dec 7 '18 at 15:25
add a comment |
$begingroup$
Example: $M=N=mathbb{R}$ with the standard metric. Then $operatorname{Isom}mathbb{E}^2$ is a 3-dimensional Lie group, so obviously cannot be the Cartesian square of $operatorname{Isom}mathbb{E}^1$.
answered Dec 6 '18 at 8:44
user10354138user10354138
7,4322925
$endgroup$
$begingroup$
Thank you for your answer. What can be say in compact case?
$endgroup$
– C.F.G
Dec 7 '18 at 5:00
1
$begingroup$
For example, see the link given by @AnthonyCarapetis
$endgroup$
– user10354138
Dec 7 '18 at 15:25
add a comment |
$begingroup$
Example: $M=N=mathbb{R}$ with the standard metric. Then $operatorname{Isom}mathbb{E}^2$ is a 3-dimensional Lie group, so obviously cannot be the Cartesian square of $operatorname{Isom}mathbb{E}^1$.
answered Dec 6 '18 at 8:44
user10354138user10354138
7,4322925
$endgroup$
Example: $M=N=mathbb{R}$ with the standard metric. Then $operatorname{Isom}mathbb{E}^2$ is a 3-dimensional Lie group, so obviously cannot be the Cartesian square of $operatorname{Isom}mathbb{E}^1$.
answered Dec 6 '18 at 8:44
user10354138user10354138
7,4322925
answered Dec 6 '18 at 8:44
user10354138user10354138
7,4322925
answered Dec 6 '18 at 8:44
user10354138user10354138
7,4322925
answered Dec 6 '18 at 8:44
user10354138user10354138
7,4322925
7,4322925
$begingroup$
Thank you for your answer. What can be say in compact case?
$endgroup$
– C.F.G
Dec 7 '18 at 5:00
1
$begingroup$
For example, see the link given by @AnthonyCarapetis
$endgroup$
– user10354138
Dec 7 '18 at 15:25
add a comment |
$begingroup$
Thank you for your answer. What can be say in compact case?
$endgroup$
– C.F.G
Dec 7 '18 at 5:00
1
$begingroup$
For example, see the link given by @AnthonyCarapetis
$endgroup$
– user10354138
Dec 7 '18 at 15:25
$begingroup$
Thank you for your answer. What can be say in compact case?
$endgroup$
– C.F.G
Dec 7 '18 at 5:00
$begingroup$
Thank you for your answer. What can be say in compact case?
$endgroup$
– C.F.G
Dec 7 '18 at 5:00
1
1
$begingroup$
For example, see the link given by @AnthonyCarapetis
$endgroup$
– user10354138
Dec 7 '18 at 15:25
$begingroup$
For example, see the link given by @AnthonyCarapetis
$endgroup$
– user10354138
Dec 7 '18 at 15:25
add a comment |
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$begingroup$
What do you mean by "for product metrics"? $M$ and $N$ should both be non-trivial Riemannian products of non-flat factors?
$endgroup$
– Anthony Carapetis
Dec 6 '18 at 9:42
1
$begingroup$
This seems relevant.
$endgroup$
– Anthony Carapetis
Dec 6 '18 at 9:48