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- Is there any research done on conservation of (integrated over all spacetime) spacetime curvature?

After trying to kinda get a picture of the field of play in quantum physics according to the standard model, a question came up. I tried to formulate the known bosons each as a particle transferring some property.

1. Photons transfer electric charge: the electromagnetic force gives attraction between positive and negative charges so an electronic current

2. Gluons transfer color charge

3. The 3 weak bosons transfer weak isospin

4. The Higgs mechanism transfers a combination of weak isospin and weak hypercharge

5. Gravitons, if they exist, transfer spacetime curvature?

The first 4 properties transferred are conserved (by quantum conservation laws). My question is: has any research been done on whether spacetime curvature (integrated over all of it) could be conserved? I'm especially interested in loop quantum gravity views on this matter (hope I can sort of understand that but OK)

The only thing I found on internet is a physics stackexchange question https://physics.stackexchange.com/questions/130379/conservation-in-space-time-curvature where the question is waved away with the observation that the universe at large is not curved, so what curvature? But IMO that might be in fact an indication that every negative curvature induces an equal opposite positive curvature somehow.

1. Photons transfer electric charge: the electromagnetic force gives attraction between positive and negative charges so an electronic current

2. Gluons transfer color charge

3. The 3 weak bosons transfer weak isospin

4. The Higgs mechanism transfers a combination of weak isospin and weak hypercharge

5. Gravitons, if they exist, transfer spacetime curvature?

The first 4 properties transferred are conserved (by quantum conservation laws). My question is: has any research been done on whether spacetime curvature (integrated over all of it) could be conserved? I'm especially interested in loop quantum gravity views on this matter (hope I can sort of understand that but OK)

The only thing I found on internet is a physics stackexchange question https://physics.stackexchange.com/questions/130379/conservation-in-space-time-curvature where the question is waved away with the observation that the universe at large is not curved, so what curvature? But IMO that might be in fact an indication that every negative curvature induces an equal opposite positive curvature somehow.