Buzz Bloom said:
I am OK using your usage, which matches Wikipedia.: an n-sphere is an n-dimensional volume in an n-dimensional Euclidean space where all points of this volume are less than or equal to a specified distance from a point specified as the center.
No, that is not my usage. The correct term for what you are quoting here from Wikipedia is an n-
ball, not an n-sphere. (More precisely, a closed n-ball, since it says "less than or equal to"; if it were just "less than" it would be an open ball.)
A common definition of an n-sphere is that it is the n-dimensional boundary of an n+1-ball. However, that requires the n-sphere to be embedded in a higher dimensional space. That is not the case for our universe. So to have a definition that makes sense for our universe, we need to avoid anything that requires an embedding.
Buzz Bloom said:
I am assuming that a 3D finite universe can be considered to be the collection of all points at a specified distance from a center point of a 4-dimentional hyper-sphere.
This definition requires an embedding, which we need to avoid. See above. For a "no embedding" definition, see the end of this post.
Buzz Bloom said:
If I am understanding your nomenclature correctly
You're not. See above.
Buzz Bloom said:
the quote above seems to be describing a 3D space with a 2D boundary.
No, it isn't. It's describing a 3D space with a finite volume but no boundary.
Buzz Bloom said:
Surely a finite universe also has no boundaries.
Correct. See above.
To come up with a "no embedding" definition, first consider the case of a 2-sphere. This is a 2-dimensional surface with a finite area but no boundary. Its area can be expressed as ##4 \pi r^2##, where ##r## is usually called the "radius of curvature" of the 2-sphere. But we have to be careful, because "radius" naturally suggests an embedding in 3-dimensional Euclidean space, which is what we want to avoid. However, there is an alternative choice: ##r## is the Gaussian curvature of the 2-sphere, i.e., its intrinsic curvature, considered as a manifold in its own right. We can measure this intrinsic curvature by measuring geodesic deviation on the 2-sphere. (In GR terms, the Riemann tensor for a 2-dimensional manifold has only one independent component, the Gaussian curvature of the surface, which in the general case can be a function of the coordinates: but if the one component is positive and constant, i.e., independent of the coordinates, then we have a 2-sphere, and the one constant independent component is ##r##.)
The "no embedding" definition of the 3-sphere is similar: the only difference is that now the 3-volume of the 3-sphere is ##2 \pi^2 r^3##, where ##r## is the constant Gaussian (intrinsic) curvature of the 3-sphere. Here the connection with the Riemann tensor is a bit more complicated, because the Riemann tensor for a 3-dimensional manifold in the general case has 6 independent components; but for a 3-sphere those 6 components are all equal, and the Gaussian curvature ##r## is the Ricci scalar derived from that Riemann tensor.
