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 MRN is a connected smooth (C) submanifoldMathworldPlanetmath and VRN is a real analytic subvariety of the same dimension as M, such that MV. Then M is a real analytic submanifold.

The condition that M is smooth cannot be relaxed to Ck for k<. For example, note that in 2, the subvariety y3-x8=0, which is the graph of the C1 function y=|x|83, 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