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TrickyDicky

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The FRW cosmology is a solution of the FRW that can be foliated into 3D isotropic and homogeneous slices.

This foliation is implemented mathematically first by the use of a not generally covariant coordinate condition https://en.wikipedia.org/wiki/Coordinate_conditions#Synchronous_coordinates , namely the use of synchronous coordinates https://en.wikipedia.org/wiki/Synchronous_coordinates, with a synchronous time coordinate and drdt crossterms excluded .This coordinate condition is necessary to define a scale factor for the spacelike part and maximally symmetric(isotropic and homogeneous) spatial slices.

This is explained in any cosmology or GR's cosmology section textbook.

The question that I think has a simple enough answer is, from the above, isn't it obvious mathematically that the isotropicity and homogeneity properties of those 3D slices are

This foliation is implemented mathematically first by the use of a not generally covariant coordinate condition https://en.wikipedia.org/wiki/Coordinate_conditions#Synchronous_coordinates , namely the use of synchronous coordinates https://en.wikipedia.org/wiki/Synchronous_coordinates, with a synchronous time coordinate and drdt crossterms excluded .This coordinate condition is necessary to define a scale factor for the spacelike part and maximally symmetric(isotropic and homogeneous) spatial slices.

This is explained in any cosmology or GR's cosmology section textbook.

The question that I think has a simple enough answer is, from the above, isn't it obvious mathematically that the isotropicity and homogeneity properties of those 3D slices are

**not in general**(not in general since in certain specific cases like for instance in flat spacetime or in de Sitter spacetime they are indeed geometric features of the 4-manifold) geometric features for the 4D spacetime given that they come imposed by the coordinate condition.
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