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Thank You

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- Thread starter superbat
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- #1

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Thank You

- #2

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I think it's a loose way of saying that in everyday operations of differential geometry, and in particular of general relativity, one can derive any piece of info we want about curvature from the Riemann tensor without having to take additional derivatives (which would take the order of differentiation from two to at least three) or taking fancy tensor products (which the Weyl tensor requires).

- #3

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Since you mention metric, I was also wondering what does contravariant version of metric mean.

Covariant version of metric tells us about distance between 2 points. What does contravariant version of metric physically mean?

- #4

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Contravariant version has components ##g_{ab}##. It is a linear function that takes two covectors (aka one-forms or dual vectors) in the cotangent space at the relevant manifold point as input and returns a real scalar as output.

- #5

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So do they have any physical significance or should I consider them as just mathematical tools

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