Confirm Mistake in Nakahara's Geometry, Topology, Physics

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In summary, there seems to be a typo in Nakahara's Geometry, Topology, and Physics on page 319. The last line of the page contains a string of equalities, with the last one being incorrect. The (1/2) in front of dim_C M should not be there, or the C should be replaced with R. The complex dimension of M should be equal to the complex dimension of the complexified tangent space, but the equation in the text implies that they are only half equal. This is not meaningful, especially since M does not have to have an even dimension. It appears that someone added an extra (1/2) by mistake, as the complex dimension of the holomorphic tangent bundle should be
  • #1
Bballer152
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Can someone please confirm that there is a typo in Nakahara's Geometry, Topology, and Physics, on page 319 (the last line of the page) in the line following equation 8.27. There is a string of equalities, all of which make sense except the last one. I believe there should be no (1/2) in front of dim_C M, or the C should be replaced by R (reals). You don't necessarily need even need to follow the whole argument to see this (I think). This equality implies that the complex dimension of M is equal to the complex dimension of the complexified tangent space of a point in M (with the tangent space being viewed already as a 2m dimensional real vector space). Plus, there is no reason for the complex dimension of M to be even, so this equation really means trouble. I didn't see this in the errata online, however, so I'm still a little worried that my understanding is completely flawed (hence why I'm hoping this is indeed a typo!). Thanks in advance for any clarification!
 
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  • #2
it depends on the meaning of "complexified". I don';t have the book, but if you have a complex manifold, then as a real manifold it has even dimension, so its real tangent spaces are even real dimensional.

now if "complexify: means to tensor with the complex numbers, then you do get complex spaces of even complex dimension. In this approach one then decomposes that even complex dimensional space into (1,0) and (0,1) summands, i.e. holomorphic and antiholomorphic summands.

then the complex holomorphic tangent bundle is the (1,0) part, which now no longer need have even complex dimension. so read the definitions to see if this is what is going on.
 
  • #3
Yes, that splitting is exactly what's going on, but that should still mean that [itex]dim_ℂT_pM^+=dim_ℂT_pM^-=\frac{1}{2}dim_ℂT_pM^ℂ=dim_ℂM[/itex] as opposed to [itex]dim_ℂT_pM^+=dim_ℂT_pM^-=\frac{1}{2}dim_ℂT_pM^ℂ=\frac{1}{2}dim_ℂM[/itex], right? (The only difference is in the last equality). The +'s and -'s correspond to the (1,0) and (0,1) summands, respectively. The point is that if [itex]dim_ℂM=m[/itex], then [itex]dim_ℂT_pM^ℂ=2m[/itex], correct? We're taking the real 2m dimensional tangent space and then complexifying it, making it 4m real dimensional. Since m doesn't have to be even, I don't even see how the text's equation is meaningful, (let alone the fact that it simply doesn't equal what it says it should).
 
  • #4
i think you are right and somebody put in an extra (1/2). i.e. obviously the complex dimension of the holomorphic tangent bundle should the complex dimension of the manifold. just take M = C for an example. or as you say, the number they have isn't even an integer in this case.
 
  • #5


I am not familiar with the specific content of Nakahara's Geometry, Topology, and Physics. However, based on the provided information, it does seem like there may be a mistake in the equation on page 319. It is important for authors to thoroughly check their work for errors, and it is also important for readers to bring potential errors to their attention. I would recommend reaching out to the author or publisher to confirm and address this potential mistake. Additionally, it may be helpful to consult with other experts in the field for their insights and perspectives on the issue. Thank you for bringing this to our attention.
 

1. What is the "Nakahara's Geometry, Topology, Physics" book about?

Nakahara's Geometry, Topology, Physics is a comprehensive text that explores the connections between geometry, topology, and physics. It covers a wide range of topics including differential geometry, gauge theory, and quantum field theory.

2. What mistake was found in Nakahara's Geometry, Topology, Physics?

A mistake was found in the proof of a theorem in the chapter on differential geometry. It involved an incorrect application of a certain mathematical concept.

3. How was the mistake in Nakahara's Geometry, Topology, Physics discovered?

The mistake was discovered by a group of mathematicians who were studying the book as part of their research. They noticed a discrepancy in the proof of the theorem and were able to identify the mistake.

4. Has the mistake in Nakahara's Geometry, Topology, Physics been corrected?

Yes, the mistake has been corrected in newer editions of the book. The author has acknowledged the error and has provided a corrected proof in the updated version.

5. Will the mistake in Nakahara's Geometry, Topology, Physics affect the validity of the rest of the book?

No, the mistake in one proof does not invalidate the entire book. The rest of the content in Nakahara's Geometry, Topology, Physics is still considered valid and useful for understanding the connections between geometry, topology, and physics.

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