In his elegant 2012 introduction to smooth manifolds, Nigel Hitchen minimizes his reliance on charts. In stating what it means for a manifold $M$ to be a smooth submanifold of $N$, for example, he gives the condintion that $D\iota_x$ is injective for all $x\in M$. For me, at least, it would be easier to digest a condition that for any smooth function on $N$ its restriction to $M$ is smooth, but am I correct in inferring that the two conditions are equivalent?
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As a good reference for embedded vs immersed submanifolds check the book of Frank Warner -- Foundations of Differentiable Manifolds and Lie Groups. where a lot of subtleties are discussed.
orangeskid
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