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 analytic subvariety 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, note that 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.

Title	smooth submanifold contained in a subvariety of same dimension is real analytic
Canonical name	SmoothSubmanifoldContainedInASubvarietyOfSameDimensionIsRealAnalytic
Date of creation	2013-03-22 17:41:16
Last modified on	2013-03-22 17:41:16
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	5
Author	jirka (4157)
Entry type	Theorem
Classification	msc 14P99
Related topic	RealAnalyticSubvariety