Does Proposition 5.18 also hold if $N$ is changed to a smooth manifold with boundary?
2026-03-25 19:11:21.1774465881
Does Proposition 5.18 also hold if $N$ is changed to a smooth manifold with boundary?
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Since this excerpt is from my book Introduction to Smooth Manifolds, I guess I'm the one who should answer the question.
Yes, this proposition holds if $N$ has nonempty boundary, with exactly the same proof. I didn't state it that way for two reasons: (1) submanifolds with boundary are not defined until later in the chapter (page 120), and (2) as I remarked just before Proposition 5.49, "Many (though not all) of the earlier results in this chapter have analogues for submanifolds with boundary. Since we will have little reason to consider nonembedded submanifolds with boundary, we focus primarily on the embedded case."