I have a hyper-surface that globally is a 3-sphere $S^3$, but locally has an additional curvature. Reduced by one dimension it would look like a surface of a round planet with hills and valleys. The curvature of the hyper-surface (the "hyper-hills" and "hyper-valleys") can be described as needed by the appropriate tensor, etc.

In a general case, this curved hyper-surface cannot be embedded in a flat Euclidean space $X={\Bbb R}^4$, because the curvature can have more degrees of freedom, I assume. What condition would restrict the curvature to always allow the described isometric embedding? Any help or direction from the experts would be of a great value. Thank you!


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