Two real algebraic manifolds are equivalent iff they are analytically homeomorphic (Nash 1952).

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.

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Weisstein, Eric W. "Nash's Embedding Theorem." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/NashsEmbeddingTheorem.html