I know of ways to prove that a given subset of a smooth manifold is a smooth submanifold, but what if I have some subset which I suspect is not a smooth submanifold? What are some approaches to proving that it is not?
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2It depends on your definition of what a submanifold is... – Mariano Suárez-Álvarez Aug 09 '13 at 20:06
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What leads you to suspect otherwise? That might suggest something (or perhaps not). For example, maybe you suspect that there is a "corner"; then you could prove somewhere there can't be a smooth parametrization, etc. – Christopher A. Wong Aug 09 '13 at 20:47
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If $S\subset M$ is a subset of the manifold $M$, then it has a tangent cone $C(s)$ at every $s\in S$, which consists of the tangent vectors at $0$ of smooth curves $\gamma:I\to S$, where $ I\subset \mathbb R$ is an open interval containing $0 $ .
A necessary condition for $S$ to be a submanifold of dimension $k$ is that $C(s)$ be a vector space of dimension $k$ at every $s\in S$.
An example
If $M=\mathbb R^2$ and $S$ is the graph of the function $|x|$, then at $O=(0,0)$ the cone $C(O)$ is the zero vector space and at every other point of $s\in S$ the tangent cone $C(s)$ is $1$-dimensional.
Thus $S$ is not a submanifold of $M$.
Georges Elencwajg
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