The tangent bundle of a fibre bundle associated to a principal $G$-bundle.
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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
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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
4,32011225
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
4,32011225
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
4,32011225
asked Nov 21 at 11:26
Tyrone
4,32011225
asked Nov 21 at 11:26
Tyrone
4,32011225
asked Nov 21 at 11:26
Tyrone
4,32011225
asked Nov 21 at 11:26
Tyrone
4,32011225
4,32011225
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