Covariant and exterior derivatives

  • Thread starter Rasalhague
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Is Penrose implying in the bit you highlighted that he will actually be defining the numbers [itex]\alpha_{r...t}[/itex] differently from "some people", or is he just saying that there are two conventions for writing the same quantity, one of which (his) is simpler to write but involves some redundancy? (In #24 I assumed the latter.) Are "other people"'s components of k-forms equal to 1/k! times Penrose's components? If so, does this fully account for the different definitions of the exterior derivative, and to what set of tensors does this difference in convention apply (e.g. forms only, or all antisymmetric tensors, the the antisymmetric part of all tensors, or...)?

Is Penrose's definition equivalent to that of Kobayashi/Nomizu, assuming no torsion in either case? Would it possible, in principle, to deduce this, given the information I have, or would I need to know more about what conventions Kobayashi/Nomizu have adopted.

Do the Cristoffel symbols vanish in these definitions for some reason related to them being symmetric in their lower indices?
 
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Yes, it seems his conventions are the same as in Kobayashi. His "components" are different from those of "some people". Here is an extract from p. 97 of Fecko, "Differential Geometry and Lie Groups for Physicists" (very good book):
Fecko_p97.jpg
 
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Argh, so antisymmetrisation, the wedge product, the exterior derivative, and even the components of forms (and possibly other tensors) each suffer from differing conventions in how they're defined?! Thanks for the warning! I think the best thing for me to do is to try and get the hang of one consistent set of definitions, then branch out from there. Guessing (or trying to "derive") what one author means from my partial, and likely flawed, understanding of another author's system is probably not the most efficient way to learn the basic concepts. But it certainly helps to be aware that there are differences, and where they lie. As with any complex suject, comparing different explanations can often clarify a point.
 
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I think the best thing for me to do is to try and get the hang of one consistent set of definitions, then branch out from there.
Yes, this is the best solution. Personally, I did not choose the convention of Kobayashi. My choice was that of "some people".
 

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