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Thanks, Mark

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In summary, the theorem can be used to calculate the general curvature of a spacetime, but it does not always apply to pseudo-Riemannian manifolds and the manifold must be compact.f

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Thanks, Mark

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What do you mean by "amount of gravity"? Without specifying this, your question cannot be answered.

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but is gravity conserved?

you mean the total mass? I suggest to reformulate the question...

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I think I am right that the total amount of color charge and electric charge in the Universe is conserved, but is gravity conserved?

"Gravity" is not a charge like color charge and electric charge. The closest analogue to a charge for gravity would be energy. As for whether that is conserved, read this:

http://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/

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In that sense, could one not define some concept of "total curvature" ( in the context of GR ) that is in fact a conserved quantity ? This is easily visualised by taking a sphere - you can twist, distort and deform that sphere in any which way you want, but no matter what you do, you will never be able to deform it into a flat plane, i.e. you will never be able to completely eliminate the curvature that was intrinsic to the manifold when you started. The same should be true for gravity - you cannot locally "eliminate" it, the most you can do is spread it out as gravitational radiation.

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I am thinking of the ( generalised ) Gauss-Bonnet theorem

AFAIK this theorem only applies to Riemannian manifolds, not pseudo-Riemannian, so it would not apply to spacetime.

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AFAIK this theorem only applies to Riemannian manifolds, not pseudo-Riemannian, so it would not apply to spacetime.

You are right, I overlooked that However, it seems that the theorem can be extended to cover pseudo-Riemannian manifolds as well, so long as they are orientable :

https://duetosymmetry.com/files/An. Acad. Brasil. Ci. 1963 Chern.pdf

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it seems that the theorem can be extended to cover pseudo-Riemannian manifolds as well, so long as they are orientable

Thanks for the link, interesting!

However, there is also another key restriction on the theorem, which also appears to apply to the pseudo-Riemannian version: the manifold must be compact. Unfortunately, that is not true of any spacetime of physical interest that I'm aware of. The closest possibility would be a closed FRW universe, for which each spacelike slice of constant comoving time is compact; but even there the spacetime as a whole, as a 4-d manifold, is not compact. So unfortunately I don't think there is any way to use this theorem in the way you're proposing.

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That's what happens when an amateur's enthusiasm for the subject outpaces his limited knowledge !

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