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I supposed the definition of the sphere was : "the locus of points of a surface at equal distance from the center", then the 3-sphere was just that the points of a higher dimensional space at equal distance from a center ?You said in Post #1"

Then obviously it's a 3 sphere and verifying the metric is euclidean is not so hard.

Since the metric of this 3 sphere is euclidean..."

1) The manifold is not a 3 sphere.

2) A 3 sphere cannot have a Euclidean metric.