smooth submanifold contained in a subvariety of same dimension is real analytic

This theorem seems to usually be attributed to Malgrange in literature as it appeared in his book[1].

Theorem (Malgrange).

Suppose $M\\subset{\\mathbb{R}}^{N}$ is a connected smooth ( $C^{\\infty}$ ) submanifold and $V\\subset{\\mathbb{R}}^{N}$ is a real analyticsubvariety of the same dimension as $M$ , such that $M\\subset V$ . Then $M$ is a real analytic submanifold.

The condition that $M$ is smooth cannot be relaxed to $C^{k}$ for $k<\\infty$ . For example, notethat in ${\\mathbb{R}}^{2}$ , the subvariety $y^{3}-x^{8}=0$ , which is the graph of the $C^{1}$ function $y=\\lvert x\\rvert^{\\frac{8}{3}}$ , is not a real analytic submanifold.

References

1 Bernard Malgrange..Oxford University Press, 1966.

单词	SmoothSubmanifoldContainedInASubvarietyOfSameDimensionIsRealAnalytic
释义	smooth submanifold contained in a subvariety of same dimension is real analytic This theorem seems to usually be attributed to Malgrange in literature as it appeared in his book[1]. Theorem (Malgrange). Suppose $M\\subset{\\mathbb{R}}^{N}$ is a connected smooth ( $C^{\\infty}$ ) submanifold and $V\\subset{\\mathbb{R}}^{N}$ is a real analyticsubvariety of the same dimension as $M$ , such that $M\\subset V$ . Then $M$ is a real analytic submanifold. The condition that $M$ is smooth cannot be relaxed to $C^{k}$ for $k<\\infty$ . For example, notethat in ${\\mathbb{R}}^{2}$ , the subvariety $y^{3}-x^{8}=0$ , which is the graph of the $C^{1}$ function $y=\\lvert x\\rvert^{\\frac{8}{3}}$ , is not a real analytic submanifold. References 1 Bernard Malgrange..Oxford University Press, 1966.
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