# 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 SmoothSubmanifoldContainedInASubvarietyOfSameDimensionIsRealAnalytic 2013-03-22 17:41:16 2013-03-22 17:41:16 jirka (4157) jirka (4157) 5 jirka (4157) Theorem msc 14P99 RealAnalyticSubvariety