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

The condition that $M$ is smooth cannot be relaxed to ${C}^{k}$ for $$. For example, note that in ${\mathbb{R}}^{2}$, the subvariety ${y}^{3}-{x}^{8}=0$, which is the graph of the ${C}^{1}$ function $y={|x|}^{\frac{8}{3}}$, 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 |