As space is isotropic which conservation law is obtained
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Answer:
It is a little different in General Relativity. Let's start with Special Relativity and all the 3 forces of the Standard Model in physics. Then we will talk about gravity and the universe.
In The Standard Model spacetime is Minkowski, meaning flat in all 4 dimensions. If it is that way clearly any direction and position is equivalent. That's called rotational symmetry and translation symmetry. No position or direction is favored, there is no reason to. Yes, from that you can deduce that linear and angular momentum are conserved. Time symmetry tells you that energy is conserved. All 3 of the standard model forces involve interactions high satisfy these symmetries. And some others.
But the universe is a very large system that also includes gravity. With gravity the spacetime can be bent, or generally have curvature. Isotropy and homogeneity now depends on how that gravity, caused by matter and energy, evolved. Sure microscopically the symmetries of special relativity hold (rotation= isotropy, translation= homogeneity),and General Relativity reduces to Special Relativity in local regions near the observer. But when we go to the big, such as the universe, it is General Relativity with a more statistical or thermodynamic representation of the matter and energy.
Thus, whether the universe is homogeneous or isotopic is not now a given. You must observe the universe. We have been observing the stars and galaxies, and in the large (i.e., for distances on the order of a couple megaparsecs or more) the distribution of matter we see in the universe seems to be the same anywhere we look (at those large aggregations of objects). They look the same in all directions circularly, and at all distances linearly. Rotationally and translationally invariant, in the large.
We have more proof of that from the cosmic microwave background, microwave radio waves which is the redshifted products of the light and radiation produced about 380000 years after the Big Bang. We have detected it and it is extremely homogeneous and isotropy. We think it has to be produced that symmetrically then.
In General Relativity that leads to only one possible family of solutions to the equations. It's a Robertson Walker Friedman universe. We got all its equations, and it all agrees with all our measurements of how fast the universe expands, and everything else (to a high degree of precision, but not 100%).
In General Relativity we go the other way. Spacetime is homogeneous and isotropic, so we conclude that momentum and angular momentum is conserved. So interactions in the universe, in the large, conserve these entities.
The next question you asked was why is spacetime isotropic and homogeneous. Well, this was also due to the way the univesrse evolved. But it is known that if it started that way, with the only non symmetries in those being due to quantum fluctuations, the epochs of time when the matter and energy were in contact they had to be in thermal equilibrium, and thermalized each other to a similar symmetric distribution, plus or minus the fluctuations that grew (density fluctuation tended to grow because of gravity's attraction, forming galaxies and stars and planets). But in the large the symmetries remained. The story is more complex, with another important part contributed by the so called space inflation, a huge growth in a very short time, which actually tended to then have galaxies separated over longer distances than the light could have traveled still be in equilibrium so that the symmetries are seen in the very large distances. The story of galaxies and stars evolving is also pretty complex, but the principle simple: gravity brought matter to coalesce (often in very violent collisions etc)
So the symmetries and conservation: yes, but more complex and the universe homogeneity and isotropy came from the Big Bang and thermalization, along with inflation.
And in General Relativity since the universe is not time symmetric, energy as understood in it is not conserved. Some people say that gravity and matter/energy somehow balance out that energy between them. Not too important an issue, since it is known how the interactions happen and how to account for them.