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Question about differential geometry 
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#1
Feb813, 09:50 AM

P: 11

Hi, I read in Padmanabhan's book that [itex]\nabla_a J^a=0[/itex] implies that there exists an antisymetric tensor P such that [itex]J^a= \nabla_b P^{ba}[/itex]. What's the name of the theorem? Any reference?
Thanks 


#2
Feb813, 09:54 AM

P: 2,785

having trouble reading your latex rendering. Can you fix it?



#3
Feb813, 10:34 AM

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P: 1,945




#4
Feb813, 11:02 AM

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P: 11,895

Question about differential geometry



#5
Feb813, 11:16 AM

C. Spirit
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P: 5,427




#6
Feb813, 11:22 AM

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It's not the converse. It's the direct lemma.
http://en.wikipedia.org/wiki/Poincar...ar.C3.A9_lemma Because d2 = 0, any exact form is automatically closed. The question of whether every closed form is exact depends on the topology of the domain of interest. On a contractible domain, every closed form is exact by the Poincaré lemma. More general questions of this kind on an arbitrary differentiable manifold are the subject of de Rham cohomology, that allows one to obtain purely topological information using differential methods. 


#7
Feb813, 11:33 AM

C. Spirit
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The result [itex]\partial _{a }v^{a} = 0 \Rightarrow \exists P^{ab} = P^{ba}:v^{a}=\partial _{b}P^{ab}[/itex] is gotten by applying a consequence of the poincare lemma. The result needed by the OP comes from the fact that if [itex]d\alpha = 0[/itex] then locally [itex]\exists \beta :\alpha = d\beta [/itex]. Maybe converse wasn't the word to use here if that's what you are saying dexter; Wald does use that word but Lee doesn't so I don't know what to say other than that Wald might not be using it in the logical sense but rather in an informal/literal sense of the word (EDIT: Lee proves it and Wald doesn't  for the OP's interest)



#8
Feb813, 11:41 AM

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P: 1,945

The converses in the 3D case are:
I think that these cases follow from Gauss' theorem and 


#9
Feb813, 05:05 PM

Mentor
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#10
Feb913, 03:16 AM

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P: 888

Some weeks ago there was a question about this also here. You can check Tomas Ortin's book "Gravity and Strings" on this theorem, it probably states its name.



#11
Feb913, 03:29 PM

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I couldn't find it in Ortin (excellent book, btw).



#12
Feb913, 03:48 PM

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<Poincare lemma. Let U be an open ball in E and let A be a differential form of degree >= 1 on U such that dA = 0. Then there exists a differential form B on U such that dB = A.> This is the mathematical standard result picked up from <Serge Lang, Differential Manifolds, Springer Verlag, 1985>. Let's go to <Spivak, Calculus on Manifolds, AddisonWesley> Page 94: <411 Theorem (Poincare Lemma). If [itex] A\subset \mathbb{R}^{n} [/itex] is an open set starshaped with respect to 0, then every closed form on A is exact.> *From Flanders's text, his first words from this preface to the first (1963) edition (quoted by Wald). <Last spring the author gave a series of lectures on exterior differential forms to a group of faculty members and graduate students from the Purdue Engineering Schools. The material that was covered in these lectures is presented here in an expanded version. The book is aimed primarily at engineers and physical scientists in the hope of making available to them new tools of very great power in modern mathematics.> 


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