Mixing
Reduced bundles and global sections of associated bundle
$begingroup$
I am following Kobayashi and Nomizu's book Foundations of differential geometry volume 1 (page no $57$) Proposition 5.6.
The structure group of a principal bundle $P(M,G)$ is reducible to a closed subgroup $H$ of $G$ if and only if the associated bundle $E=(Ptimes G/H)/Grightarrow M$ has a global section. (Here we use the notation $Q(M,H)$ for the reduced bundle.)
One direction I was able to understand. Given that $G$ is reducible to $H$, I was able to produce a global section for $Erightarrow M$.
For the other direction, I was able to understand everything except that I could not prove $Q$ is an immersed submanifold.
If the following result is true, then I am done.
Is the inverse image of an immersed submanifold an immersed submanifold under smooth submersion?
I know the above result is true for embedded submanifold (Using transversality) but I am not sure about the result for immersed submanifold.
differential-geometry smooth-manifolds principal-bundles
asked Jan 30 at 9:25
WandereradiWandereradi
838
$endgroup$
add a comment |
$begingroup$
I am following Kobayashi and Nomizu's book Foundations of differential geometry volume 1 (page no $57$) Proposition 5.6.
The structure group of a principal bundle $P(M,G)$ is reducible to a closed subgroup $H$ of $G$ if and only if the associated bundle $E=(Ptimes G/H)/Grightarrow M$ has a global section. (Here we use the notation $Q(M,H)$ for the reduced bundle.)
One direction I was able to understand. Given that $G$ is reducible to $H$, I was able to produce a global section for $Erightarrow M$.
For the other direction, I was able to understand everything except that I could not prove $Q$ is an immersed submanifold.
If the following result is true, then I am done.
Is the inverse image of an immersed submanifold an immersed submanifold under smooth submersion?
I know the above result is true for embedded submanifold (Using transversality) but I am not sure about the result for immersed submanifold.
differential-geometry smooth-manifolds principal-bundles
asked Jan 30 at 9:25
WandereradiWandereradi
838
$endgroup$
add a comment |
$begingroup$
I am following Kobayashi and Nomizu's book Foundations of differential geometry volume 1 (page no $57$) Proposition 5.6.
The structure group of a principal bundle $P(M,G)$ is reducible to a closed subgroup $H$ of $G$ if and only if the associated bundle $E=(Ptimes G/H)/Grightarrow M$ has a global section. (Here we use the notation $Q(M,H)$ for the reduced bundle.)
One direction I was able to understand. Given that $G$ is reducible to $H$, I was able to produce a global section for $Erightarrow M$.
For the other direction, I was able to understand everything except that I could not prove $Q$ is an immersed submanifold.
If the following result is true, then I am done.
Is the inverse image of an immersed submanifold an immersed submanifold under smooth submersion?
I know the above result is true for embedded submanifold (Using transversality) but I am not sure about the result for immersed submanifold.
differential-geometry smooth-manifolds principal-bundles
asked Jan 30 at 9:25
WandereradiWandereradi
838
$endgroup$
I am following Kobayashi and Nomizu's book Foundations of differential geometry volume 1 (page no $57$) Proposition 5.6.
The structure group of a principal bundle $P(M,G)$ is reducible to a closed subgroup $H$ of $G$ if and only if the associated bundle $E=(Ptimes G/H)/Grightarrow M$ has a global section. (Here we use the notation $Q(M,H)$ for the reduced bundle.)
One direction I was able to understand. Given that $G$ is reducible to $H$, I was able to produce a global section for $Erightarrow M$.
For the other direction, I was able to understand everything except that I could not prove $Q$ is an immersed submanifold.
If the following result is true, then I am done.
Is the inverse image of an immersed submanifold an immersed submanifold under smooth submersion?
I know the above result is true for embedded submanifold (Using transversality) but I am not sure about the result for immersed submanifold.
differential-geometry smooth-manifolds principal-bundles
differential-geometry smooth-manifolds principal-bundles
asked Jan 30 at 9:25
WandereradiWandereradi
838
asked Jan 30 at 9:25
WandereradiWandereradi
838
edited Jan 30 at 11:26
Wandereradi
asked Jan 30 at 9:25
WandereradiWandereradi
838
asked Jan 30 at 9:25
WandereradiWandereradi
838
asked Jan 30 at 9:25
WandereradiWandereradi
838
838
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Yes - use that an immersion is locally an embedding to reduce to the case of an embedded submanifold.
Added:
Let $U$ be the immersed submanifold with preimage $V$. Cover $U$ by embedded submanifolds $U_i$. The preimages $V_i$ are embedded and cover $V$, so $V$ is an immersion.
answered Jan 30 at 14:15
BenBen
4,293617
$endgroup$
$begingroup$
Thanks @Ben for the answer!..I think you are talking about local embedding theorem. But to use the result to boil it down to the embedded case I am needing another result that I am not sure of.. I need union of embedded submanifolds is an embedded submanifold... Is that true? Or we have to approach in a different way? I am Sorry if my remark is stupid.
$endgroup$
– Wandereradi
Jan 30 at 14:36
$begingroup$
@ADITTYACHAUDHURI The union of embedded manifolds is only an immersed manifold. But that good enough right?
$endgroup$
– Ben
Jan 30 at 15:51
$begingroup$
@ADITTYACHAUDHURI By the way I added another line of explanation.
$endgroup$
– Ben
Jan 30 at 17:53
$begingroup$
@ Ben Thanks! Actually I never came across about the result that the union of embedded submanifiolds is an immersed submanifold. Since there are examples where union of submanifolds is not a submanifold I never thought in that direction.Can you please refer me a literature (or give a brief outline ) for the proof of the fact that union of embedded submanifolds is an immersed submanifold? I could not see directly that local embedding theorem is equivalent to "This Result". Sorry in advance if my remark is stupid!
$endgroup$
– Wandereradi
Jan 31 at 1:07
1
$begingroup$
@ADITTYACHAUDHURI Here by "union of embedded submanifolds" I just mean it is a union of open subsets which are embedded. That is, given $M to N$ and $U_i$ covering $M$ such that $U_i to N$ are embeddings, I am saying "$M$ is a union of embedded manifolds". But this is just the condition of locally being an embedding.
$endgroup$
– Ben
Jan 31 at 5:44
|
show 1 more comment
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1 Answer
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active
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votes
1 Answer
1
active
oldest
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active
oldest
votes
$begingroup$
Yes - use that an immersion is locally an embedding to reduce to the case of an embedded submanifold.
Added:
Let $U$ be the immersed submanifold with preimage $V$. Cover $U$ by embedded submanifolds $U_i$. The preimages $V_i$ are embedded and cover $V$, so $V$ is an immersion.
answered Jan 30 at 14:15
BenBen
4,293617
$endgroup$
$begingroup$
Thanks @Ben for the answer!..I think you are talking about local embedding theorem. But to use the result to boil it down to the embedded case I am needing another result that I am not sure of.. I need union of embedded submanifolds is an embedded submanifold... Is that true? Or we have to approach in a different way? I am Sorry if my remark is stupid.
$endgroup$
– Wandereradi
Jan 30 at 14:36
$begingroup$
@ADITTYACHAUDHURI The union of embedded manifolds is only an immersed manifold. But that good enough right?
$endgroup$
– Ben
Jan 30 at 15:51
$begingroup$
@ADITTYACHAUDHURI By the way I added another line of explanation.
$endgroup$
– Ben
Jan 30 at 17:53
$begingroup$
@ Ben Thanks! Actually I never came across about the result that the union of embedded submanifiolds is an immersed submanifold. Since there are examples where union of submanifolds is not a submanifold I never thought in that direction.Can you please refer me a literature (or give a brief outline ) for the proof of the fact that union of embedded submanifolds is an immersed submanifold? I could not see directly that local embedding theorem is equivalent to "This Result". Sorry in advance if my remark is stupid!
$endgroup$
– Wandereradi
Jan 31 at 1:07
1
$begingroup$
@ADITTYACHAUDHURI Here by "union of embedded submanifolds" I just mean it is a union of open subsets which are embedded. That is, given $M to N$ and $U_i$ covering $M$ such that $U_i to N$ are embeddings, I am saying "$M$ is a union of embedded manifolds". But this is just the condition of locally being an embedding.
$endgroup$
– Ben
Jan 31 at 5:44
|
show 1 more comment
$begingroup$
Yes - use that an immersion is locally an embedding to reduce to the case of an embedded submanifold.
Added:
Let $U$ be the immersed submanifold with preimage $V$. Cover $U$ by embedded submanifolds $U_i$. The preimages $V_i$ are embedded and cover $V$, so $V$ is an immersion.
answered Jan 30 at 14:15
BenBen
4,293617
$endgroup$
$begingroup$
Thanks @Ben for the answer!..I think you are talking about local embedding theorem. But to use the result to boil it down to the embedded case I am needing another result that I am not sure of.. I need union of embedded submanifolds is an embedded submanifold... Is that true? Or we have to approach in a different way? I am Sorry if my remark is stupid.
$endgroup$
– Wandereradi
Jan 30 at 14:36
$begingroup$
@ADITTYACHAUDHURI The union of embedded manifolds is only an immersed manifold. But that good enough right?
$endgroup$
– Ben
Jan 30 at 15:51
$begingroup$
@ADITTYACHAUDHURI By the way I added another line of explanation.
$endgroup$
– Ben
Jan 30 at 17:53
$begingroup$
@ Ben Thanks! Actually I never came across about the result that the union of embedded submanifiolds is an immersed submanifold. Since there are examples where union of submanifolds is not a submanifold I never thought in that direction.Can you please refer me a literature (or give a brief outline ) for the proof of the fact that union of embedded submanifolds is an immersed submanifold? I could not see directly that local embedding theorem is equivalent to "This Result". Sorry in advance if my remark is stupid!
$endgroup$
– Wandereradi
Jan 31 at 1:07
1
$begingroup$
@ADITTYACHAUDHURI Here by "union of embedded submanifolds" I just mean it is a union of open subsets which are embedded. That is, given $M to N$ and $U_i$ covering $M$ such that $U_i to N$ are embeddings, I am saying "$M$ is a union of embedded manifolds". But this is just the condition of locally being an embedding.
$endgroup$
– Ben
Jan 31 at 5:44
|
show 1 more comment
$begingroup$
Yes - use that an immersion is locally an embedding to reduce to the case of an embedded submanifold.
Added:
Let $U$ be the immersed submanifold with preimage $V$. Cover $U$ by embedded submanifolds $U_i$. The preimages $V_i$ are embedded and cover $V$, so $V$ is an immersion.
answered Jan 30 at 14:15
BenBen
4,293617
$endgroup$
Yes - use that an immersion is locally an embedding to reduce to the case of an embedded submanifold.
Added:
Let $U$ be the immersed submanifold with preimage $V$. Cover $U$ by embedded submanifolds $U_i$. The preimages $V_i$ are embedded and cover $V$, so $V$ is an immersion.
answered Jan 30 at 14:15
BenBen
4,293617
edited Jan 30 at 15:49
answered Jan 30 at 14:15
BenBen
4,293617
answered Jan 30 at 14:15
BenBen
4,293617
answered Jan 30 at 14:15
BenBen
4,293617
4,293617
$begingroup$
Thanks @Ben for the answer!..I think you are talking about local embedding theorem. But to use the result to boil it down to the embedded case I am needing another result that I am not sure of.. I need union of embedded submanifolds is an embedded submanifold... Is that true? Or we have to approach in a different way? I am Sorry if my remark is stupid.
$endgroup$
– Wandereradi
Jan 30 at 14:36
$begingroup$
@ADITTYACHAUDHURI The union of embedded manifolds is only an immersed manifold. But that good enough right?
$endgroup$
– Ben
Jan 30 at 15:51
$begingroup$
@ADITTYACHAUDHURI By the way I added another line of explanation.
$endgroup$
– Ben
Jan 30 at 17:53
$begingroup$
@ Ben Thanks! Actually I never came across about the result that the union of embedded submanifiolds is an immersed submanifold. Since there are examples where union of submanifolds is not a submanifold I never thought in that direction.Can you please refer me a literature (or give a brief outline ) for the proof of the fact that union of embedded submanifolds is an immersed submanifold? I could not see directly that local embedding theorem is equivalent to "This Result". Sorry in advance if my remark is stupid!
$endgroup$
– Wandereradi
Jan 31 at 1:07
1
$begingroup$
@ADITTYACHAUDHURI Here by "union of embedded submanifolds" I just mean it is a union of open subsets which are embedded. That is, given $M to N$ and $U_i$ covering $M$ such that $U_i to N$ are embeddings, I am saying "$M$ is a union of embedded manifolds". But this is just the condition of locally being an embedding.
$endgroup$
– Ben
Jan 31 at 5:44
|
show 1 more comment
$begingroup$
Thanks @Ben for the answer!..I think you are talking about local embedding theorem. But to use the result to boil it down to the embedded case I am needing another result that I am not sure of.. I need union of embedded submanifolds is an embedded submanifold... Is that true? Or we have to approach in a different way? I am Sorry if my remark is stupid.
$endgroup$
– Wandereradi
Jan 30 at 14:36
$begingroup$
@ADITTYACHAUDHURI The union of embedded manifolds is only an immersed manifold. But that good enough right?
$endgroup$
– Ben
Jan 30 at 15:51
$begingroup$
@ADITTYACHAUDHURI By the way I added another line of explanation.
$endgroup$
– Ben
Jan 30 at 17:53
$begingroup$
@ Ben Thanks! Actually I never came across about the result that the union of embedded submanifiolds is an immersed submanifold. Since there are examples where union of submanifolds is not a submanifold I never thought in that direction.Can you please refer me a literature (or give a brief outline ) for the proof of the fact that union of embedded submanifolds is an immersed submanifold? I could not see directly that local embedding theorem is equivalent to "This Result". Sorry in advance if my remark is stupid!
$endgroup$
– Wandereradi
Jan 31 at 1:07
1
$begingroup$
@ADITTYACHAUDHURI Here by "union of embedded submanifolds" I just mean it is a union of open subsets which are embedded. That is, given $M to N$ and $U_i$ covering $M$ such that $U_i to N$ are embeddings, I am saying "$M$ is a union of embedded manifolds". But this is just the condition of locally being an embedding.
$endgroup$
– Ben
Jan 31 at 5:44
$begingroup$
Thanks @Ben for the answer!..I think you are talking about local embedding theorem. But to use the result to boil it down to the embedded case I am needing another result that I am not sure of.. I need union of embedded submanifolds is an embedded submanifold... Is that true? Or we have to approach in a different way? I am Sorry if my remark is stupid.
$endgroup$
– Wandereradi
Jan 30 at 14:36
$begingroup$
Thanks @Ben for the answer!..I think you are talking about local embedding theorem. But to use the result to boil it down to the embedded case I am needing another result that I am not sure of.. I need union of embedded submanifolds is an embedded submanifold... Is that true? Or we have to approach in a different way? I am Sorry if my remark is stupid.
$endgroup$
– Wandereradi
Jan 30 at 14:36
$begingroup$
@ADITTYACHAUDHURI The union of embedded manifolds is only an immersed manifold. But that good enough right?
$endgroup$
– Ben
Jan 30 at 15:51
$begingroup$
@ADITTYACHAUDHURI The union of embedded manifolds is only an immersed manifold. But that good enough right?
$endgroup$
– Ben
Jan 30 at 15:51
$begingroup$
@ADITTYACHAUDHURI By the way I added another line of explanation.
$endgroup$
– Ben
Jan 30 at 17:53
$begingroup$
@ADITTYACHAUDHURI By the way I added another line of explanation.
$endgroup$
– Ben
Jan 30 at 17:53
$begingroup$
@ Ben Thanks! Actually I never came across about the result that the union of embedded submanifiolds is an immersed submanifold. Since there are examples where union of submanifolds is not a submanifold I never thought in that direction.Can you please refer me a literature (or give a brief outline ) for the proof of the fact that union of embedded submanifolds is an immersed submanifold? I could not see directly that local embedding theorem is equivalent to "This Result". Sorry in advance if my remark is stupid!
$endgroup$
– Wandereradi
Jan 31 at 1:07
$begingroup$
@ Ben Thanks! Actually I never came across about the result that the union of embedded submanifiolds is an immersed submanifold. Since there are examples where union of submanifolds is not a submanifold I never thought in that direction.Can you please refer me a literature (or give a brief outline ) for the proof of the fact that union of embedded submanifolds is an immersed submanifold? I could not see directly that local embedding theorem is equivalent to "This Result". Sorry in advance if my remark is stupid!
$endgroup$
– Wandereradi
Jan 31 at 1:07
1
1
$begingroup$
@ADITTYACHAUDHURI Here by "union of embedded submanifolds" I just mean it is a union of open subsets which are embedded. That is, given $M to N$ and $U_i$ covering $M$ such that $U_i to N$ are embeddings, I am saying "$M$ is a union of embedded manifolds". But this is just the condition of locally being an embedding.
$endgroup$
– Ben
Jan 31 at 5:44
$begingroup$
@ADITTYACHAUDHURI Here by "union of embedded submanifolds" I just mean it is a union of open subsets which are embedded. That is, given $M to N$ and $U_i$ covering $M$ such that $U_i to N$ are embeddings, I am saying "$M$ is a union of embedded manifolds". But this is just the condition of locally being an embedding.
$endgroup$
– Ben
Jan 31 at 5:44
|
show 1 more comment
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