Two real algebraic manifolds are equivalent iff they are analytically homeomorphic (Nash 1952).
Nash's Embedding Theorem
See also
EmbeddingExplore with Wolfram|Alpha
References
Kowalczyk, A. "Whitney's and Nash's Embedding Theorems for Differential Spaces." Bull. Acad. Polon. Sci. Sér. Sci. Math. 28, 385-390, 1981.Masahiro, S. Nash Manifolds. Berlin: Springer-Verlag, 1987.Nash, J. "Real Algebraic Manifolds." Ann. Math. 56, 405-421, 1952.Referenced on Wolfram|Alpha
Nash's Embedding TheoremCite this as:
Weisstein, Eric W. "Nash's Embedding Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/NashsEmbeddingTheorem.html