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Pictures below are from do Carmo's Riemannian Geometry.

$f:M\rightarrow \overline M$ is immersion. When codimension is 1, according to the book, there is $\nabla^\perp_X \eta=0$. But I need an extra condition $|\eta|=1$ to prove $\nabla^\perp_X \eta=0$.

What I do: since $|\eta|=1$, I have $$ 0=X\langle \eta,\eta\rangle=2\langle \overline \nabla_X \eta, \eta\rangle $$ since codimension is 1, the normal component of $\overline \nabla_X \eta$ is zero. Namely $\nabla ^\perp_X \eta=0$.

But without $|\eta|=1$, I don't know how to prove it. In fact, I feel do Carmo miss the $|\eta|=1$, but I am not sure, so ask here.

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Enhao Lan
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    Advice: It is not appropriate to upload most of pages of a book in this website. – C.F.G Sep 15 '21 at 12:40
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    Yes you need to assume $\lvert\eta\rvert$ is constant along integral curves of $X$, or else you pick up some $(X\log \lvert\eta\rvert)\eta$. – user10354138 Sep 15 '21 at 12:51
  • @C.F.G Thanks for your reminding. I have edit it. Is it suitable? I am not sure the reason of not appropriate to upload of most of pages. Is it copyright? Or too much to read ? – Enhao Lan Sep 16 '21 at 01:49
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    @lanse7pty: I just worry about possible future problems (penalty, ...) for Math.SE and not other things. – C.F.G Sep 16 '21 at 03:36

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