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What manifold structure is natural for the codomain of a differential form defined on a manifold?
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It is well known that a differential form $omega$, of degree $r$, defined in a manifold $M^m$ is an application that at each point $pin M^m $ associates an alternating $r$-linear form $omega(p)in bigwedge^{r}(T_pM^m)$.
I understand that codomain of the differential form $omega$ would then be $bigcup_{pin M} bigwedge^{r}(T_pM^m)$.
Question. What manifold structure is natural for the codomain $bigcup_{pin M} bigwedge^{r}(T_pM^m)$ of a differential form $omega$ defined on a manifold $M^m$?
By "natural" I mean that if the form $omega$ is a $ C^k $ form, then the application
$$
omega: Mto bigcup_{pin M} bigwedge^{r}(T_pM^m)
$$
is a $C^k$ application between the manifolds $M$ and $bigcup_{pin M} bigwedge^{r}(T_pM^m)$.
smooth-manifolds differential-forms multilinear-algebra smooth-functions
asked Feb 2 at 12:49
MathOverviewMathOverview
8,94743164
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add a comment |
$begingroup$
It is well known that a differential form $omega$, of degree $r$, defined in a manifold $M^m$ is an application that at each point $pin M^m $ associates an alternating $r$-linear form $omega(p)in bigwedge^{r}(T_pM^m)$.
I understand that codomain of the differential form $omega$ would then be $bigcup_{pin M} bigwedge^{r}(T_pM^m)$.
Question. What manifold structure is natural for the codomain $bigcup_{pin M} bigwedge^{r}(T_pM^m)$ of a differential form $omega$ defined on a manifold $M^m$?
By "natural" I mean that if the form $omega$ is a $ C^k $ form, then the application
$$
omega: Mto bigcup_{pin M} bigwedge^{r}(T_pM^m)
$$
is a $C^k$ application between the manifolds $M$ and $bigcup_{pin M} bigwedge^{r}(T_pM^m)$.
smooth-manifolds differential-forms multilinear-algebra smooth-functions
asked Feb 2 at 12:49
MathOverviewMathOverview
8,94743164
$endgroup$
$begingroup$
Are you taking about the tangent bundle?
$endgroup$
– Peter Saveliev
Feb 2 at 16:54
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@PeterSaveliev Sorry, I do not understand what your questioning is. Could you explain better?
$endgroup$
– MathOverview
Feb 2 at 17:00
1
$begingroup$
en.wikipedia.org/wiki/Tangent_bundle
$endgroup$
– Peter Saveliev
Feb 2 at 18:58
add a comment |
$begingroup$
It is well known that a differential form $omega$, of degree $r$, defined in a manifold $M^m$ is an application that at each point $pin M^m $ associates an alternating $r$-linear form $omega(p)in bigwedge^{r}(T_pM^m)$.
I understand that codomain of the differential form $omega$ would then be $bigcup_{pin M} bigwedge^{r}(T_pM^m)$.
Question. What manifold structure is natural for the codomain $bigcup_{pin M} bigwedge^{r}(T_pM^m)$ of a differential form $omega$ defined on a manifold $M^m$?
By "natural" I mean that if the form $omega$ is a $ C^k $ form, then the application
$$
omega: Mto bigcup_{pin M} bigwedge^{r}(T_pM^m)
$$
is a $C^k$ application between the manifolds $M$ and $bigcup_{pin M} bigwedge^{r}(T_pM^m)$.
smooth-manifolds differential-forms multilinear-algebra smooth-functions
asked Feb 2 at 12:49
MathOverviewMathOverview
8,94743164
$endgroup$
It is well known that a differential form $omega$, of degree $r$, defined in a manifold $M^m$ is an application that at each point $pin M^m $ associates an alternating $r$-linear form $omega(p)in bigwedge^{r}(T_pM^m)$.
I understand that codomain of the differential form $omega$ would then be $bigcup_{pin M} bigwedge^{r}(T_pM^m)$.
Question. What manifold structure is natural for the codomain $bigcup_{pin M} bigwedge^{r}(T_pM^m)$ of a differential form $omega$ defined on a manifold $M^m$?
By "natural" I mean that if the form $omega$ is a $ C^k $ form, then the application
$$
omega: Mto bigcup_{pin M} bigwedge^{r}(T_pM^m)
$$
is a $C^k$ application between the manifolds $M$ and $bigcup_{pin M} bigwedge^{r}(T_pM^m)$.
smooth-manifolds differential-forms multilinear-algebra smooth-functions
smooth-manifolds differential-forms multilinear-algebra smooth-functions
asked Feb 2 at 12:49
MathOverviewMathOverview
8,94743164
asked Feb 2 at 12:49
MathOverviewMathOverview
8,94743164
asked Feb 2 at 12:49
MathOverviewMathOverview
8,94743164
asked Feb 2 at 12:49
MathOverviewMathOverview
8,94743164
asked Feb 2 at 12:49
MathOverviewMathOverview
8,94743164
8,94743164
$begingroup$
Are you taking about the tangent bundle?
$endgroup$
– Peter Saveliev
Feb 2 at 16:54
$begingroup$
@PeterSaveliev Sorry, I do not understand what your questioning is. Could you explain better?
$endgroup$
– MathOverview
Feb 2 at 17:00
1
$begingroup$
en.wikipedia.org/wiki/Tangent_bundle
$endgroup$
– Peter Saveliev
Feb 2 at 18:58
add a comment |
$begingroup$
Are you taking about the tangent bundle?
$endgroup$
– Peter Saveliev
Feb 2 at 16:54
$begingroup$
@PeterSaveliev Sorry, I do not understand what your questioning is. Could you explain better?
$endgroup$
– MathOverview
Feb 2 at 17:00
1
$begingroup$
en.wikipedia.org/wiki/Tangent_bundle
$endgroup$
– Peter Saveliev
Feb 2 at 18:58
$begingroup$
Are you taking about the tangent bundle?
$endgroup$
– Peter Saveliev
Feb 2 at 16:54
$begingroup$
Are you taking about the tangent bundle?
$endgroup$
– Peter Saveliev
Feb 2 at 16:54
$begingroup$
@PeterSaveliev Sorry, I do not understand what your questioning is. Could you explain better?
$endgroup$
– MathOverview
Feb 2 at 17:00
$begingroup$
@PeterSaveliev Sorry, I do not understand what your questioning is. Could you explain better?
$endgroup$
– MathOverview
Feb 2 at 17:00
1
1
$begingroup$
en.wikipedia.org/wiki/Tangent_bundle
$endgroup$
– Peter Saveliev
Feb 2 at 18:58
$begingroup$
en.wikipedia.org/wiki/Tangent_bundle
$endgroup$
– Peter Saveliev
Feb 2 at 18:58
add a comment |
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$begingroup$
Are you taking about the tangent bundle?
$endgroup$
– Peter Saveliev
Feb 2 at 16:54
$begingroup$
@PeterSaveliev Sorry, I do not understand what your questioning is. Could you explain better?
$endgroup$
– MathOverview
Feb 2 at 17:00
1
$begingroup$
en.wikipedia.org/wiki/Tangent_bundle
$endgroup$
– Peter Saveliev
Feb 2 at 18:58