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Motivation for construction of Associated fiber bundle from a principal bundle
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Given a principal $G$ bundle $P(M,G)$ and a manifold $F$ with an action of $G$ on it from left, we construct a fibre bundle over $M$ with fiber $F$ and call this the associated fiber bundle for $P(M,G)$.
I do not get the motivation behind the construction given in Kobayashi and Nomizu which I will write down below.
Idea is to construct a fibre bundle with fibre $F$ i.e., we need to construct a smooth manifold $E$ and a smooth map $pi_E:Erightarrow M$ that gives a fiber bundle with fiber $F$.
Kobayashi's proof goes as follows :
They consider the product manifold $Ptimes F$ with an action of $G$ as $g.(u,xi)=(ug,g^-1xi)$. Then they consider the quotient space $(Ptimes F)/G$ and call this $E$.
Consider the map projection map $Ptimes Frightarrow M$ defined as $(p,xi)mapsto pi(p)$.
This induces a map $pi_E:E=(Ptimes F)/Grightarrow M$. As $Prightarrow M$ is a principal $G$ bundle given $xin M$ there exists an open set $U$ containing $x$ and a local trivialization $pi^-1(U)rightarrow Utimes G$. They then give a bijection $pi_E^-1(U)rightarrow Utimes F$ and give a smooth structure on $E$ so that these bijections are difeomorphisms. Then, they cal $(E,pi_E,M,P,F)$ the fiber bundle associated to principal $G$ bundle.
I am trying to understand the motivation for the above construction.
Suppose $F=H$, a Lie group and the action of $G$ on $H$ is given by a morphism of Lie grroup $phi:Grightarrow H$ with $Gtimes Hrightarrow H$ given by $(g,h)mapsto phi(g)^-1h$ do get a principal $H$ bundle in above construction?
dg.differential-geometry lie-groups principal-bundles
asked 4 hours ago
Praphulla Koushik
5811321
 |Â
show 1 more comment
up vote
2
down vote
favorite
Given a principal $G$ bundle $P(M,G)$ and a manifold $F$ with an action of $G$ on it from left, we construct a fibre bundle over $M$ with fiber $F$ and call this the associated fiber bundle for $P(M,G)$.
I do not get the motivation behind the construction given in Kobayashi and Nomizu which I will write down below.
Idea is to construct a fibre bundle with fibre $F$ i.e., we need to construct a smooth manifold $E$ and a smooth map $pi_E:Erightarrow M$ that gives a fiber bundle with fiber $F$.
Kobayashi's proof goes as follows :
They consider the product manifold $Ptimes F$ with an action of $G$ as $g.(u,xi)=(ug,g^-1xi)$. Then they consider the quotient space $(Ptimes F)/G$ and call this $E$.
Consider the map projection map $Ptimes Frightarrow M$ defined as $(p,xi)mapsto pi(p)$.
This induces a map $pi_E:E=(Ptimes F)/Grightarrow M$. As $Prightarrow M$ is a principal $G$ bundle given $xin M$ there exists an open set $U$ containing $x$ and a local trivialization $pi^-1(U)rightarrow Utimes G$. They then give a bijection $pi_E^-1(U)rightarrow Utimes F$ and give a smooth structure on $E$ so that these bijections are difeomorphisms. Then, they cal $(E,pi_E,M,P,F)$ the fiber bundle associated to principal $G$ bundle.
I am trying to understand the motivation for the above construction.
Suppose $F=H$, a Lie group and the action of $G$ on $H$ is given by a morphism of Lie grroup $phi:Grightarrow H$ with $Gtimes Hrightarrow H$ given by $(g,h)mapsto phi(g)^-1h$ do get a principal $H$ bundle in above construction?
dg.differential-geometry lie-groups principal-bundles
asked 4 hours ago
Praphulla Koushik
5811321
$G$ does not act on its subgroups in any reasonable way; I guess you wanted to turn that arrow around. In that case, yes. But I think the most important application of this construction is the definition of associated vector bundle to a principal bundle given a representation of $G$.
– Mike Miller
4 hours ago
@MikeMiller I have edited the content... Can you say something on motivation behind such definition... As you said, given a representation $Grightarrow Gl(V)$ which is same thing as a smooth action of $G$ on $V$ we can talk about associated fiber bundle.. Fiber in this case is vector space so we have a associated vector bundle in this case..
– Praphulla Koushik
3 hours ago
Is your question about potential applications of associated bundles as a motivation for studying them? If yes, you may want to look at gauge theories in physics.
– S.Surace
2 hours ago
@S.Surace I don’t understand your statement “Is your question about potential applications of associated bundles as a motivation for studying them?†I want to know the motivation behind this particular construction of associated fiber bundles
– Praphulla Koushik
2 hours ago
It seems that you have other potential constructions of associated bundles in mind. The one you are citing is the standard one and there are many examples of smooth manifolds that can be viewed from this angle. Moreover, the construction happens to coincide with what is needed for gauge theories in physics. The idea is to consistently "attach" the gauge symmetry group $G$ as well as all the matter fields that transform under this group to the base manifold $M$ and at the same time identifying configurations that are related by a gauge transformation.
– S.Surace
1 hour ago
 |Â
show 1 more comment
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Given a principal $G$ bundle $P(M,G)$ and a manifold $F$ with an action of $G$ on it from left, we construct a fibre bundle over $M$ with fiber $F$ and call this the associated fiber bundle for $P(M,G)$.
I do not get the motivation behind the construction given in Kobayashi and Nomizu which I will write down below.
Idea is to construct a fibre bundle with fibre $F$ i.e., we need to construct a smooth manifold $E$ and a smooth map $pi_E:Erightarrow M$ that gives a fiber bundle with fiber $F$.
Kobayashi's proof goes as follows :
They consider the product manifold $Ptimes F$ with an action of $G$ as $g.(u,xi)=(ug,g^-1xi)$. Then they consider the quotient space $(Ptimes F)/G$ and call this $E$.
Consider the map projection map $Ptimes Frightarrow M$ defined as $(p,xi)mapsto pi(p)$.
This induces a map $pi_E:E=(Ptimes F)/Grightarrow M$. As $Prightarrow M$ is a principal $G$ bundle given $xin M$ there exists an open set $U$ containing $x$ and a local trivialization $pi^-1(U)rightarrow Utimes G$. They then give a bijection $pi_E^-1(U)rightarrow Utimes F$ and give a smooth structure on $E$ so that these bijections are difeomorphisms. Then, they cal $(E,pi_E,M,P,F)$ the fiber bundle associated to principal $G$ bundle.
I am trying to understand the motivation for the above construction.
Suppose $F=H$, a Lie group and the action of $G$ on $H$ is given by a morphism of Lie grroup $phi:Grightarrow H$ with $Gtimes Hrightarrow H$ given by $(g,h)mapsto phi(g)^-1h$ do get a principal $H$ bundle in above construction?
dg.differential-geometry lie-groups principal-bundles
asked 4 hours ago
Praphulla Koushik
5811321
Given a principal $G$ bundle $P(M,G)$ and a manifold $F$ with an action of $G$ on it from left, we construct a fibre bundle over $M$ with fiber $F$ and call this the associated fiber bundle for $P(M,G)$.
I do not get the motivation behind the construction given in Kobayashi and Nomizu which I will write down below.
Idea is to construct a fibre bundle with fibre $F$ i.e., we need to construct a smooth manifold $E$ and a smooth map $pi_E:Erightarrow M$ that gives a fiber bundle with fiber $F$.
Kobayashi's proof goes as follows :
They consider the product manifold $Ptimes F$ with an action of $G$ as $g.(u,xi)=(ug,g^-1xi)$. Then they consider the quotient space $(Ptimes F)/G$ and call this $E$.
Consider the map projection map $Ptimes Frightarrow M$ defined as $(p,xi)mapsto pi(p)$.
This induces a map $pi_E:E=(Ptimes F)/Grightarrow M$. As $Prightarrow M$ is a principal $G$ bundle given $xin M$ there exists an open set $U$ containing $x$ and a local trivialization $pi^-1(U)rightarrow Utimes G$. They then give a bijection $pi_E^-1(U)rightarrow Utimes F$ and give a smooth structure on $E$ so that these bijections are difeomorphisms. Then, they cal $(E,pi_E,M,P,F)$ the fiber bundle associated to principal $G$ bundle.
I am trying to understand the motivation for the above construction.
Suppose $F=H$, a Lie group and the action of $G$ on $H$ is given by a morphism of Lie grroup $phi:Grightarrow H$ with $Gtimes Hrightarrow H$ given by $(g,h)mapsto phi(g)^-1h$ do get a principal $H$ bundle in above construction?
dg.differential-geometry lie-groups principal-bundles
dg.differential-geometry lie-groups principal-bundles
asked 4 hours ago
Praphulla Koushik
5811321
asked 4 hours ago
Praphulla Koushik
5811321
edited 3 hours ago
asked 4 hours ago
Praphulla Koushik
5811321
asked 4 hours ago
Praphulla Koushik
5811321
asked 4 hours ago
Praphulla Koushik
5811321
5811321
$G$ does not act on its subgroups in any reasonable way; I guess you wanted to turn that arrow around. In that case, yes. But I think the most important application of this construction is the definition of associated vector bundle to a principal bundle given a representation of $G$.
– Mike Miller
4 hours ago
@MikeMiller I have edited the content... Can you say something on motivation behind such definition... As you said, given a representation $Grightarrow Gl(V)$ which is same thing as a smooth action of $G$ on $V$ we can talk about associated fiber bundle.. Fiber in this case is vector space so we have a associated vector bundle in this case..
– Praphulla Koushik
3 hours ago
Is your question about potential applications of associated bundles as a motivation for studying them? If yes, you may want to look at gauge theories in physics.
– S.Surace
2 hours ago
@S.Surace I don’t understand your statement “Is your question about potential applications of associated bundles as a motivation for studying them?†I want to know the motivation behind this particular construction of associated fiber bundles
– Praphulla Koushik
2 hours ago
It seems that you have other potential constructions of associated bundles in mind. The one you are citing is the standard one and there are many examples of smooth manifolds that can be viewed from this angle. Moreover, the construction happens to coincide with what is needed for gauge theories in physics. The idea is to consistently "attach" the gauge symmetry group $G$ as well as all the matter fields that transform under this group to the base manifold $M$ and at the same time identifying configurations that are related by a gauge transformation.
– S.Surace
1 hour ago
 |Â
show 1 more comment
$G$ does not act on its subgroups in any reasonable way; I guess you wanted to turn that arrow around. In that case, yes. But I think the most important application of this construction is the definition of associated vector bundle to a principal bundle given a representation of $G$.
– Mike Miller
4 hours ago
@MikeMiller I have edited the content... Can you say something on motivation behind such definition... As you said, given a representation $Grightarrow Gl(V)$ which is same thing as a smooth action of $G$ on $V$ we can talk about associated fiber bundle.. Fiber in this case is vector space so we have a associated vector bundle in this case..
– Praphulla Koushik
3 hours ago
Is your question about potential applications of associated bundles as a motivation for studying them? If yes, you may want to look at gauge theories in physics.
– S.Surace
2 hours ago
@S.Surace I don’t understand your statement “Is your question about potential applications of associated bundles as a motivation for studying them?†I want to know the motivation behind this particular construction of associated fiber bundles
– Praphulla Koushik
2 hours ago
It seems that you have other potential constructions of associated bundles in mind. The one you are citing is the standard one and there are many examples of smooth manifolds that can be viewed from this angle. Moreover, the construction happens to coincide with what is needed for gauge theories in physics. The idea is to consistently "attach" the gauge symmetry group $G$ as well as all the matter fields that transform under this group to the base manifold $M$ and at the same time identifying configurations that are related by a gauge transformation.
– S.Surace
1 hour ago
$G$ does not act on its subgroups in any reasonable way; I guess you wanted to turn that arrow around. In that case, yes. But I think the most important application of this construction is the definition of associated vector bundle to a principal bundle given a representation of $G$.
– Mike Miller
4 hours ago
$G$ does not act on its subgroups in any reasonable way; I guess you wanted to turn that arrow around. In that case, yes. But I think the most important application of this construction is the definition of associated vector bundle to a principal bundle given a representation of $G$.
– Mike Miller
4 hours ago
@MikeMiller I have edited the content... Can you say something on motivation behind such definition... As you said, given a representation $Grightarrow Gl(V)$ which is same thing as a smooth action of $G$ on $V$ we can talk about associated fiber bundle.. Fiber in this case is vector space so we have a associated vector bundle in this case..
– Praphulla Koushik
3 hours ago
@MikeMiller I have edited the content... Can you say something on motivation behind such definition... As you said, given a representation $Grightarrow Gl(V)$ which is same thing as a smooth action of $G$ on $V$ we can talk about associated fiber bundle.. Fiber in this case is vector space so we have a associated vector bundle in this case..
– Praphulla Koushik
3 hours ago
Is your question about potential applications of associated bundles as a motivation for studying them? If yes, you may want to look at gauge theories in physics.
– S.Surace
2 hours ago
Is your question about potential applications of associated bundles as a motivation for studying them? If yes, you may want to look at gauge theories in physics.
– S.Surace
2 hours ago
@S.Surace I don’t understand your statement “Is your question about potential applications of associated bundles as a motivation for studying them?†I want to know the motivation behind this particular construction of associated fiber bundles
– Praphulla Koushik
2 hours ago
@S.Surace I don’t understand your statement “Is your question about potential applications of associated bundles as a motivation for studying them?†I want to know the motivation behind this particular construction of associated fiber bundles
– Praphulla Koushik
2 hours ago
It seems that you have other potential constructions of associated bundles in mind. The one you are citing is the standard one and there are many examples of smooth manifolds that can be viewed from this angle. Moreover, the construction happens to coincide with what is needed for gauge theories in physics. The idea is to consistently "attach" the gauge symmetry group $G$ as well as all the matter fields that transform under this group to the base manifold $M$ and at the same time identifying configurations that are related by a gauge transformation.
– S.Surace
1 hour ago
It seems that you have other potential constructions of associated bundles in mind. The one you are citing is the standard one and there are many examples of smooth manifolds that can be viewed from this angle. Moreover, the construction happens to coincide with what is needed for gauge theories in physics. The idea is to consistently "attach" the gauge symmetry group $G$ as well as all the matter fields that transform under this group to the base manifold $M$ and at the same time identifying configurations that are related by a gauge transformation.
– S.Surace
1 hour ago
 |Â
show 1 more comment
1 Answer
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To answer your specific question, you get a principal $H$-bundle, often denoted $P times^G H$. The transition maps are clearly just given by applying $phi$ to those of $P$.
I think you want a motivation for the general notion of relating principal and fiber bundles through associated bundles. Tensor bundles come up rather naturally, as do projectivized cotangent bundles, projectivized tangent bundles and sphere bundles (i.e. quotients of nonzero vectors by positive scaling) on the tangent and cotangent bundles. These bundles might appear to be beasts whose natural habitats are rather different parts of the forest, but they all live in the zoo of bundles associated to the one principal bundle: the frame bundle, i.e. the set of linear isomorphisms between a fixed vector space and the tangent spaces of our manifold. In this way, topological obstructions to the existence of linearly independent sections become characteristic classes on the one principal bundle. The method of the moving frame makes use of the existence of invariant differential forms on the frame bundle, and its various subbundles. G-structures are just subbundles of the frame bundle, but together they describe many of the most important geometric structures, and the theory of G-structures makes uniform construction of the local invariants of all of those structures. So the principal bundle is unifying object bringing all of those associated fiber bundles and vector bundles together, for topology and also for local differential geometry.
answered 2 hours ago
Ben McKay
13.3k22755
I understand your first paragraph... second paragraph is completely some other foreign language for me.. :( I understand almost nothing there.. please consider using some more words in explaining...
– Praphulla Koushik
2 hours ago
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
5
down vote
To answer your specific question, you get a principal $H$-bundle, often denoted $P times^G H$. The transition maps are clearly just given by applying $phi$ to those of $P$.
I think you want a motivation for the general notion of relating principal and fiber bundles through associated bundles. Tensor bundles come up rather naturally, as do projectivized cotangent bundles, projectivized tangent bundles and sphere bundles (i.e. quotients of nonzero vectors by positive scaling) on the tangent and cotangent bundles. These bundles might appear to be beasts whose natural habitats are rather different parts of the forest, but they all live in the zoo of bundles associated to the one principal bundle: the frame bundle, i.e. the set of linear isomorphisms between a fixed vector space and the tangent spaces of our manifold. In this way, topological obstructions to the existence of linearly independent sections become characteristic classes on the one principal bundle. The method of the moving frame makes use of the existence of invariant differential forms on the frame bundle, and its various subbundles. G-structures are just subbundles of the frame bundle, but together they describe many of the most important geometric structures, and the theory of G-structures makes uniform construction of the local invariants of all of those structures. So the principal bundle is unifying object bringing all of those associated fiber bundles and vector bundles together, for topology and also for local differential geometry.
answered 2 hours ago
Ben McKay
13.3k22755
I understand your first paragraph... second paragraph is completely some other foreign language for me.. :( I understand almost nothing there.. please consider using some more words in explaining...
– Praphulla Koushik
2 hours ago
add a comment |Â
up vote
5
down vote
To answer your specific question, you get a principal $H$-bundle, often denoted $P times^G H$. The transition maps are clearly just given by applying $phi$ to those of $P$.
I think you want a motivation for the general notion of relating principal and fiber bundles through associated bundles. Tensor bundles come up rather naturally, as do projectivized cotangent bundles, projectivized tangent bundles and sphere bundles (i.e. quotients of nonzero vectors by positive scaling) on the tangent and cotangent bundles. These bundles might appear to be beasts whose natural habitats are rather different parts of the forest, but they all live in the zoo of bundles associated to the one principal bundle: the frame bundle, i.e. the set of linear isomorphisms between a fixed vector space and the tangent spaces of our manifold. In this way, topological obstructions to the existence of linearly independent sections become characteristic classes on the one principal bundle. The method of the moving frame makes use of the existence of invariant differential forms on the frame bundle, and its various subbundles. G-structures are just subbundles of the frame bundle, but together they describe many of the most important geometric structures, and the theory of G-structures makes uniform construction of the local invariants of all of those structures. So the principal bundle is unifying object bringing all of those associated fiber bundles and vector bundles together, for topology and also for local differential geometry.
answered 2 hours ago
Ben McKay
13.3k22755
I understand your first paragraph... second paragraph is completely some other foreign language for me.. :( I understand almost nothing there.. please consider using some more words in explaining...
– Praphulla Koushik
2 hours ago
add a comment |Â
up vote
5
down vote
up vote
5
down vote
To answer your specific question, you get a principal $H$-bundle, often denoted $P times^G H$. The transition maps are clearly just given by applying $phi$ to those of $P$.
I think you want a motivation for the general notion of relating principal and fiber bundles through associated bundles. Tensor bundles come up rather naturally, as do projectivized cotangent bundles, projectivized tangent bundles and sphere bundles (i.e. quotients of nonzero vectors by positive scaling) on the tangent and cotangent bundles. These bundles might appear to be beasts whose natural habitats are rather different parts of the forest, but they all live in the zoo of bundles associated to the one principal bundle: the frame bundle, i.e. the set of linear isomorphisms between a fixed vector space and the tangent spaces of our manifold. In this way, topological obstructions to the existence of linearly independent sections become characteristic classes on the one principal bundle. The method of the moving frame makes use of the existence of invariant differential forms on the frame bundle, and its various subbundles. G-structures are just subbundles of the frame bundle, but together they describe many of the most important geometric structures, and the theory of G-structures makes uniform construction of the local invariants of all of those structures. So the principal bundle is unifying object bringing all of those associated fiber bundles and vector bundles together, for topology and also for local differential geometry.
answered 2 hours ago
Ben McKay
13.3k22755
To answer your specific question, you get a principal $H$-bundle, often denoted $P times^G H$. The transition maps are clearly just given by applying $phi$ to those of $P$.
I think you want a motivation for the general notion of relating principal and fiber bundles through associated bundles. Tensor bundles come up rather naturally, as do projectivized cotangent bundles, projectivized tangent bundles and sphere bundles (i.e. quotients of nonzero vectors by positive scaling) on the tangent and cotangent bundles. These bundles might appear to be beasts whose natural habitats are rather different parts of the forest, but they all live in the zoo of bundles associated to the one principal bundle: the frame bundle, i.e. the set of linear isomorphisms between a fixed vector space and the tangent spaces of our manifold. In this way, topological obstructions to the existence of linearly independent sections become characteristic classes on the one principal bundle. The method of the moving frame makes use of the existence of invariant differential forms on the frame bundle, and its various subbundles. G-structures are just subbundles of the frame bundle, but together they describe many of the most important geometric structures, and the theory of G-structures makes uniform construction of the local invariants of all of those structures. So the principal bundle is unifying object bringing all of those associated fiber bundles and vector bundles together, for topology and also for local differential geometry.
answered 2 hours ago
Ben McKay
13.3k22755
edited 2 hours ago
answered 2 hours ago
Ben McKay
13.3k22755
answered 2 hours ago
Ben McKay
13.3k22755
answered 2 hours ago
Ben McKay
13.3k22755
13.3k22755
I understand your first paragraph... second paragraph is completely some other foreign language for me.. :( I understand almost nothing there.. please consider using some more words in explaining...
– Praphulla Koushik
2 hours ago
add a comment |Â
I understand your first paragraph... second paragraph is completely some other foreign language for me.. :( I understand almost nothing there.. please consider using some more words in explaining...
– Praphulla Koushik
2 hours ago
I understand your first paragraph... second paragraph is completely some other foreign language for me.. :( I understand almost nothing there.. please consider using some more words in explaining...
– Praphulla Koushik
2 hours ago
I understand your first paragraph... second paragraph is completely some other foreign language for me.. :( I understand almost nothing there.. please consider using some more words in explaining...
– Praphulla Koushik
2 hours ago
add a comment |Â
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$G$ does not act on its subgroups in any reasonable way; I guess you wanted to turn that arrow around. In that case, yes. But I think the most important application of this construction is the definition of associated vector bundle to a principal bundle given a representation of $G$.
– Mike Miller
4 hours ago
@MikeMiller I have edited the content... Can you say something on motivation behind such definition... As you said, given a representation $Grightarrow Gl(V)$ which is same thing as a smooth action of $G$ on $V$ we can talk about associated fiber bundle.. Fiber in this case is vector space so we have a associated vector bundle in this case..
– Praphulla Koushik
3 hours ago
Is your question about potential applications of associated bundles as a motivation for studying them? If yes, you may want to look at gauge theories in physics.
– S.Surace
2 hours ago
@S.Surace I don’t understand your statement “Is your question about potential applications of associated bundles as a motivation for studying them?†I want to know the motivation behind this particular construction of associated fiber bundles
– Praphulla Koushik
2 hours ago
It seems that you have other potential constructions of associated bundles in mind. The one you are citing is the standard one and there are many examples of smooth manifolds that can be viewed from this angle. Moreover, the construction happens to coincide with what is needed for gauge theories in physics. The idea is to consistently "attach" the gauge symmetry group $G$ as well as all the matter fields that transform under this group to the base manifold $M$ and at the same time identifying configurations that are related by a gauge transformation.
– S.Surace
1 hour ago