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Ali Taghavi
science forum addict
Joined: 14 May 2005
Posts: 73
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 Posted: Thu Jun 08, 2006 8:07 am Post subject:
real analytic De Rahm cohomology
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Hello
Let M be a real analytic manifold,if the real analytic De Rahm cohomology is the same as smooth one?
namely:the quotien of Closed analytic form by Exact analytic form?
Thanks |
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Ali Taghavi
science forum addict
Joined: 14 May 2005
Posts: 73
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| Posted: Sun Jun 18, 2006 10:45 am Post subject:
Re: real analytic De Rahm cohomology
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about existence of real analytic metric:
there is a chapter in Hirsch 's book (Diff Topology)
"approximation functions",can this chapter implies "there is always an analytic metric"?
Further,could you please more explain about Hodge theorem and its relation to
the followqing question:
"The Dimension of Real analytic Derham cohomology:Analytic closed forms/analytic exact forms"
How can we use the fact that:there is a harmonic form Cohomolog to a given
closed form?Let da be a harmonic(Exact) form can we choose a harmonic
form B such that da=dB
thank you
| Quote: | If a real analytic paracompact manifold admits an
analytic riemannian
metric (I don't know if such is always the case),
then the inclusion of
the complex of analytic forms into that of smooth
forms induces a
Frechet isomorphism in cohomology by
http://www.iecn.u-nancy.fr/~gaillard/DIVERS/Hodge.Theorem/
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Pierre-Yves Gaillard
science forum beginner
Joined: 22 Jun 2005
Posts: 22
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| Posted: Mon Jun 19, 2006 11:51 am Post subject:
Re: real analytic De Rahm cohomology
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I hope the following is correct.
Let M be a paracompact real-analytic connected manifold. Equip M with an
analytic riemannian metric. (Such exist by the Morrey-Grauert Theorem.) Let
A be the complex of coclosed harmonic forms,
B be the complex of analytic forms,
C be the complex of smooth forms.
The inclusion A c B holds by regularity.
By
[1] http://www.iecn.u-nancy.fr/~gaillard/DIVERS/Hodge.Theorem/
the inclusion A c C induces an isomorphism in cohomology.
By the argument of [1] (and by regularity) the inclusion
A c B
induces an isomorphism in cohomology. |
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Ali Taghavi
science forum addict
Joined: 14 May 2005
Posts: 73
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| Posted: Mon Jun 19, 2006 5:49 pm Post subject:
Re: real analytic De Rahm cohomology
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it seems that we should be more carefull in use of "Inclusion"
Is not necessary to prove this statment?:
"A harmonic exact form is differential of a harmonic form"?
| Quote: | I hope the following is correct.
Let M be a paracompact real-analytic connected
manifold. Equip M with an
analytic riemannian metric. (Such exist by the
Morrey-Grauert Theorem.) Let
A be the complex of coclosed harmonic forms,
B be the complex of analytic forms,
C be the complex of smooth forms.
The inclusion A c B holds by regularity.
By
[1]
http://www.iecn.u-nancy.fr/~gaillard/DIVERS/Hodge.Theo
rem/
the inclusion A c C induces an isomorphism in
cohomology.
By the argument of [1] (and by regularity) the
inclusion
A c B
induces an isomorphism in cohomology.
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Ali Taghavi
science forum addict
Joined: 14 May 2005
Posts: 73
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| Posted: Tue Jun 20, 2006 11:31 am Post subject:
Re: real analytic De Rahm cohomology
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I am sorry for this meaningless question,since a harmonic form is closed
But I replace this with the following (which I think is necessary for the
argument of the main question, Do I persue an unnecessary "exacting"?!:
Let A be a real analytic Exact form, Does there exist an analytic form B
s.t dB=A
| Quote: | it seems that we should be more carefull in use of
"Inclusion"
Is not necessary to prove this statment?:
"A harmonic exact form is differential of a harmonic
form"?
I hope the following is correct.
Let M be a paracompact real-analytic connected
manifold. Equip M with an
analytic riemannian metric. (Such exist by the
Morrey-Grauert Theorem.) Let
A be the complex of coclosed harmonic forms,
B be the complex of analytic forms,
C be the complex of smooth forms.
The inclusion A c B holds by regularity.
By
[1]
http://www.iecn.u-nancy.fr/~gaillard/DIVERS/Hodge.Theo
rem/
the inclusion A c C induces an isomorphism in
cohomology.
By the argument of [1] (and by regularity) the
inclusion
A c B
induces an isomorphism in cohomology.
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Pierre-Yves Gaillard
science forum beginner
Joined: 22 Jun 2005
Posts: 22
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| Posted: Tue Jun 20, 2006 11:31 am Post subject:
Re: real analytic De Rahm cohomology
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Ali Taghavi wrote:
| Quote: | it seems that we should be more carefull in use of "Inclusion"
Is not necessary to prove this statment?:
"A harmonic exact form is differential of a harmonic form"?
This statement is clearly false, but we don't need it. |
Are you sure you didn't miss the word "coclosed"? |
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Pierre-Yves Gaillard
science forum beginner
Joined: 22 Jun 2005
Posts: 22
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| Posted: Wed Jun 21, 2006 11:50 am Post subject:
Re: real analytic De Rahm cohomology
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Ali Taghavi wrote:
| Quote: | I am sorry for this meaningless question,since a harmonic form is closed.
No! A harmonic form is NOT closed in general. |
| Quote: | But I replace this with the following (which I think is necessary for the
argument of the main question, Do I persue an unnecessary "exacting"?!:
Let A be a real analytic Exact form, Does there exist an analytic form B
s.t dB=A?
Yes! This a particular case of a stronger statement, which follows from |
the Hodge Theorem: the inclusion of the complex of analytic forms into
the de Rham complex induces in cohomology a continuous linear bijection
of a complete locally convex Hausdorff space to a Frechet. (I don't know
if this map is always open.) |
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