The tangent bundle of a fibre bundle associated to a principal $G$-bundle.
up vote
1
down vote
favorite
I imagine this is fairly elementary, but I couldn't find a good reference.
Let $G$ be a compact Lie group and $pi:Prightarrow M$ a principal $G$-bundle over a (compact) manifold $M$. Let $F$ be another (compact) manifold on which $G$ acts on the left through a smooth map $rho:Grightarrow Diff(F)$. We form the balanced product $Ptimes_{rho}F$ which admits a smooth submersion onto $M$ with fibre $F$.
Is there a nice global description of the tangent bundle $T(Ptimes_rho F)rightarrow Ptimes_rho F$?
Locally the question is pretty clear, since the local triviality of $P$ transfers to $Ptimes_rho F$, to give smooth fibrewise local trivialisations $(Ptimes_rho F)|_Ucong Utimes F$ for suitable open $Usubseteq M$. Thus locally $T(Ptimes_rho F)$ looks like $TUtimes TF$, and it seems to me that globally it should be the quotient vector bundle $TPtimes_{Trho}TF$ formed as the balanced product by the induced tangent representation $Trho$ of $G$.
differential-geometry differential-topology vector-bundles
asked Nov 21 at 11:26
Tyrone
4,14011125
add a comment |
up vote
1
down vote
favorite
I imagine this is fairly elementary, but I couldn't find a good reference.
Let $G$ be a compact Lie group and $pi:Prightarrow M$ a principal $G$-bundle over a (compact) manifold $M$. Let $F$ be another (compact) manifold on which $G$ acts on the left through a smooth map $rho:Grightarrow Diff(F)$. We form the balanced product $Ptimes_{rho}F$ which admits a smooth submersion onto $M$ with fibre $F$.
Is there a nice global description of the tangent bundle $T(Ptimes_rho F)rightarrow Ptimes_rho F$?
Locally the question is pretty clear, since the local triviality of $P$ transfers to $Ptimes_rho F$, to give smooth fibrewise local trivialisations $(Ptimes_rho F)|_Ucong Utimes F$ for suitable open $Usubseteq M$. Thus locally $T(Ptimes_rho F)$ looks like $TUtimes TF$, and it seems to me that globally it should be the quotient vector bundle $TPtimes_{Trho}TF$ formed as the balanced product by the induced tangent representation $Trho$ of $G$.
differential-geometry differential-topology vector-bundles
asked Nov 21 at 11:26
Tyrone
4,14011125
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I imagine this is fairly elementary, but I couldn't find a good reference.
Let $G$ be a compact Lie group and $pi:Prightarrow M$ a principal $G$-bundle over a (compact) manifold $M$. Let $F$ be another (compact) manifold on which $G$ acts on the left through a smooth map $rho:Grightarrow Diff(F)$. We form the balanced product $Ptimes_{rho}F$ which admits a smooth submersion onto $M$ with fibre $F$.
Is there a nice global description of the tangent bundle $T(Ptimes_rho F)rightarrow Ptimes_rho F$?
Locally the question is pretty clear, since the local triviality of $P$ transfers to $Ptimes_rho F$, to give smooth fibrewise local trivialisations $(Ptimes_rho F)|_Ucong Utimes F$ for suitable open $Usubseteq M$. Thus locally $T(Ptimes_rho F)$ looks like $TUtimes TF$, and it seems to me that globally it should be the quotient vector bundle $TPtimes_{Trho}TF$ formed as the balanced product by the induced tangent representation $Trho$ of $G$.
differential-geometry differential-topology vector-bundles
asked Nov 21 at 11:26
Tyrone
4,14011125
I imagine this is fairly elementary, but I couldn't find a good reference.
Let $G$ be a compact Lie group and $pi:Prightarrow M$ a principal $G$-bundle over a (compact) manifold $M$. Let $F$ be another (compact) manifold on which $G$ acts on the left through a smooth map $rho:Grightarrow Diff(F)$. We form the balanced product $Ptimes_{rho}F$ which admits a smooth submersion onto $M$ with fibre $F$.
Is there a nice global description of the tangent bundle $T(Ptimes_rho F)rightarrow Ptimes_rho F$?
Locally the question is pretty clear, since the local triviality of $P$ transfers to $Ptimes_rho F$, to give smooth fibrewise local trivialisations $(Ptimes_rho F)|_Ucong Utimes F$ for suitable open $Usubseteq M$. Thus locally $T(Ptimes_rho F)$ looks like $TUtimes TF$, and it seems to me that globally it should be the quotient vector bundle $TPtimes_{Trho}TF$ formed as the balanced product by the induced tangent representation $Trho$ of $G$.
differential-geometry differential-topology vector-bundles
differential-geometry differential-topology vector-bundles
asked Nov 21 at 11:26
Tyrone
4,14011125
asked Nov 21 at 11:26
Tyrone
4,14011125
asked Nov 21 at 11:26
Tyrone
4,14011125
asked Nov 21 at 11:26
Tyrone
4,14011125
asked Nov 21 at 11:26
Tyrone
4,14011125
4,14011125
add a comment |
add a comment |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3007592%2fthe-tangent-bundle-of-a-fibre-bundle-associated-to-a-principal-g-bundle%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
