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smooth submanifold contained in a subvariety of same dimension is real analytic
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(Theorem)
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This theorem seems to usually be attributed to Malgrange in literature as it appeared in his book[1].
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.
- 1
- Bernard Malgrange. Ideals of Differentiable Functions. Oxford University Press, 1966.
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"smooth submanifold contained in a subvariety of same dimension is real analytic" is owned by jirka.
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Cross-references: function, graph, subvariety, real analytic submanifold, dimension, real analytic subvariety, submanifold, smooth, connected, theorem
This is version 2 of smooth submanifold contained in a subvariety of same dimension is real analytic, born on 2007-12-12, modified 2007-12-12.
Object id is 10128, canonical name is SmoothSubmanifoldContainedInASubvarietyOfSameDimensionIsRealAnalytic.
Accessed 377 times total.
Classification:
| AMS MSC: | 14P99 (Algebraic geometry :: Real algebraic and real analytic geometry :: Miscellaneous) |
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Pending Errata and Addenda
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