- #1

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I am looking for a text which introduces these topics with the full topological structure (written by a mathematician would be better). Thanks.

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- Thread starter Ravi Mohan
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- #1

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I am looking for a text which introduces these topics with the full topological structure (written by a mathematician would be better). Thanks.

- #2

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There is a prequel by Lee: 'Smooth Manifolds', which may also be useful, depending on what notions you need to use. I have not felt the need to buy it yet but the result is that I am a little undercooked on vector field flows and Lie derivatives.

The downside of Lee's book is that it doesn't specifically address Pseudo-Riemannian manifolds, which is what are used in GR. In many cases the distinction doesn't matter. But sometimes it does and then you have to adapt Lee's proofs to the Pseudo-Riemannian case yourself. Unfortunately I don't know of any mathematical texts (ie not by physicists) that address Pseudo-Riemannian manifolds. Perhaps others can suggest some.

- #3

martinbn

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- #4

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Maybe O'Neills https://www.amazon.com/dp/0125267401/?tag=pfamazon01-20 could make some goodUnfortunately I don't know of any mathematical texts (ie not by physicists) that address Pseudo-Riemannian manifolds.

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- #5

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For instance the author defines an embedding as a map

[tex]

\Phi: \Sigma =\Sigma_n \hookrightarrow M_{n+1}

[/tex]

Now this definition raises questions such as

- What is the nature of this map (homeomorphism or diffeomorphism)? Wikipedia gives more rigorous definition of [itex]\Phi[/itex] as a homeomorphism onto its image.
- The author says that this embedding is represented by the parametric equations [itex]\Phi:x^{\alpha}(y^a)[/itex] How are these equations actually composed? I can only think as [itex]y^{\alpha}(\Phi^{-1}(\text{something with }x^a))[/itex] where [itex]y^{\alpha}, x^a[/itex] are charts of [itex]M,\Sigma[/itex] respectively.

The references mentioned in Wikipedia are https://en.wikipedia.org/wiki/Embedding#References, but I was not sure which one to follow.

I will also look in O'Neill's book on semi-Riemannian geometry. Thanks again.

- #6

martinbn

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- #7

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I understand what he is explaining and I think it should be enough to serve the purpose (which is understanding the canonical GR). But again it would be nice to have a complete formal definition at one's disposal (like the way Carroll does in his notes).

About the map [itex]\Phi[/itex], the author says

So I was wondering what is the stronger topological condition.Strictly speaking, such a map is called an (injective) immersion, while an embedding has to satisfy a slightly stronger topological condition, but since we are not concerned with global issues, and since I have not even tried to define what a manifold is (beyond the remarks in section 4.11), it would be ridiculous to worry about such things here and this is more than good enough.

- #8

martinbn

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So I was wondering what is the stronger topological condition.

The stronger topological condition is that the immersion ## f: M \rightarrow N## is an embedding if ##f ## is a homeomorphism of ##M## with ##f(M)## with the subspace topology. This is the case whenever ##M## is compact for example.

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- #10

George Jones

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You might try "3+1 Formalism in General Relativity: Bases of Numerical Relativity" by Eric Gourgoulhon,

https://www.amazon.com/dp/3642245242/?tag=pfamazon01-20

https://www.amazon.com/dp/3642245242/?tag=pfamazon01-20

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