Mixing
Local frame inducing a map of principal bundles
$begingroup$
Let $V rightarrow M$ a vector bundle. $P rightarrow M$ a principal $G$-bundle. Let $phi:G rightarrow GL(V)$ be a representation. A local section $s$ for $P$,
frame bundle for $V rightarrow M$, defines an isomorphism
$$psi_s: Omega^p(U) otimes Omega^0(V|_U) rightarrow Omega^p(U) otimes Omega^0(P times_G V|_U) $$
In this notation $Omega^0(underline{V}), $$Omega^0(P times_G V)$ de notes the sections of bundle. Hence, we have $p$ forms with values in this vector spaces respectively.
I am confused, how is defined?
EDIT: On second thought, I believe I have misunderstood the original question. We begin with a principal bundle $P rightarrow M$, so $P$ is not the frame bundle of $V rightarrow M$. Is this right?
Then problem seems almost equivalent to Proposition 4.1, page 7, where there no choice of local section involved.
The original question is from page 106 section 7.6.
differential-geometry differential-forms connections principal-bundles
asked Jan 22 at 8:58
CL.CL.
2,3142925
$endgroup$
add a comment |
$begingroup$
Let $V rightarrow M$ a vector bundle. $P rightarrow M$ a principal $G$-bundle. Let $phi:G rightarrow GL(V)$ be a representation. A local section $s$ for $P$,
frame bundle for $V rightarrow M$, defines an isomorphism
$$psi_s: Omega^p(U) otimes Omega^0(V|_U) rightarrow Omega^p(U) otimes Omega^0(P times_G V|_U) $$
In this notation $Omega^0(underline{V}), $$Omega^0(P times_G V)$ de notes the sections of bundle. Hence, we have $p$ forms with values in this vector spaces respectively.
I am confused, how is defined?
EDIT: On second thought, I believe I have misunderstood the original question. We begin with a principal bundle $P rightarrow M$, so $P$ is not the frame bundle of $V rightarrow M$. Is this right?
Then problem seems almost equivalent to Proposition 4.1, page 7, where there no choice of local section involved.
The original question is from page 106 section 7.6.
differential-geometry differential-forms connections principal-bundles
asked Jan 22 at 8:58
CL.CL.
2,3142925
$endgroup$
add a comment |
$begingroup$
Let $V rightarrow M$ a vector bundle. $P rightarrow M$ a principal $G$-bundle. Let $phi:G rightarrow GL(V)$ be a representation. A local section $s$ for $P$,
frame bundle for $V rightarrow M$, defines an isomorphism
$$psi_s: Omega^p(U) otimes Omega^0(V|_U) rightarrow Omega^p(U) otimes Omega^0(P times_G V|_U) $$
In this notation $Omega^0(underline{V}), $$Omega^0(P times_G V)$ de notes the sections of bundle. Hence, we have $p$ forms with values in this vector spaces respectively.
I am confused, how is defined?
EDIT: On second thought, I believe I have misunderstood the original question. We begin with a principal bundle $P rightarrow M$, so $P$ is not the frame bundle of $V rightarrow M$. Is this right?
Then problem seems almost equivalent to Proposition 4.1, page 7, where there no choice of local section involved.
The original question is from page 106 section 7.6.
differential-geometry differential-forms connections principal-bundles
asked Jan 22 at 8:58
CL.CL.
2,3142925
$endgroup$
Let $V rightarrow M$ a vector bundle. $P rightarrow M$ a principal $G$-bundle. Let $phi:G rightarrow GL(V)$ be a representation. A local section $s$ for $P$,
frame bundle for $V rightarrow M$, defines an isomorphism
$$psi_s: Omega^p(U) otimes Omega^0(V|_U) rightarrow Omega^p(U) otimes Omega^0(P times_G V|_U) $$
In this notation $Omega^0(underline{V}), $$Omega^0(P times_G V)$ de notes the sections of bundle. Hence, we have $p$ forms with values in this vector spaces respectively.
I am confused, how is defined?
EDIT: On second thought, I believe I have misunderstood the original question. We begin with a principal bundle $P rightarrow M$, so $P$ is not the frame bundle of $V rightarrow M$. Is this right?
Then problem seems almost equivalent to Proposition 4.1, page 7, where there no choice of local section involved.
The original question is from page 106 section 7.6.
differential-geometry differential-forms connections principal-bundles
differential-geometry differential-forms connections principal-bundles
asked Jan 22 at 8:58
CL.CL.
2,3142925
asked Jan 22 at 8:58
CL.CL.
2,3142925
edited Jan 22 at 9:50
CL.
asked Jan 22 at 8:58
CL.CL.
2,3142925
asked Jan 22 at 8:58
CL.CL.
2,3142925
asked Jan 22 at 8:58
CL.CL.
2,3142925
2,3142925
add a comment |
add a comment |
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