Nice! That serves perfectly for me. Thanks for your answers and patience.

In Cartan's 1938 Annali di Mat. paper, he shows that the normal curvature of any curve traced on the surface is constant. Does this imply that the surface is a quotient of the sphere $S^3$? I haven't found any other mention to the topology of the hypersurfaces..

The details you added are great! Unfortunately, I couldn't find what you wrote in the reference above. Could you give me another reference? I will need it for my work.

Thank you for your answer. Could you indicate a reference? One more thing: these hyper surfaces of $S^4$ are spheres or quotients of spheres?

Say, in diameter.