General relativity tells us that the universe is bent by gravity, but this curvature is intrinsic to the universe (the universe bends, but not in a fourth spatial dimension, the universe having only three bent spatial dimensions).
How could we know the difference between the intrinsic curvature of a 3D spatial universe, and a the extrinsic curvature of a 3D shape in a 4D universe?
Obviously, there is a similar question about our curved 4D spacetime inside a 5D spacetime.
Answer
The answer is in the "inverse square law". Gravity would follow inverse cube law if it was using the 4th dimension for curving. Gravity only knows three dimensions. In a 5 dimensional universe, gravity would follow inverse fourth power law and so on.
As inverse square law remains valid in "most realistic scenarios", it indicates that there are no more than 3 spatial dimensions in universe.
Therefore any curving has to be intrinsic.
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