TITLE: Intuitive idea of tangent space and definition coincide?
QUESTION [1 upvotes]: Let $M$ be a submanifold of $\mathbb{R}^n$ of codimension 1. Suppose you take $V\le \mathbb{R}^n$ a vector space of dimension $n-1$ and let $w \in \mathbb{R}^n\setminus\{0\}$ be an orthogonal vector to $V$.
Suppose also that, for $t \in \mathbb{R}$ sufficiently big, we have $(V+tw)\cap M = \emptyset$ and that there is a greatest $t_0 \in \mathbb{R}$ such that $(V+t_0w)\cap M \neq \emptyset$, say, $p \in  (V+t_0w)\cap M$. (For example, if $M$ is compact, this is always satisfied for any $V$.)
Is it true that $T_pM = V$? How to prove that?

REPLY [0 votes]: Suppose $w$ is of unit length otherwise we can replace it by $w/\|w\|$. Define a smooth function $f: M \rightarrow R$ by $f(q)=\langle q, w \rangle$ the inner product with $w$. We have $f(p)=t_0$ and $t_0$ is an extreme of $f$. So $p$ is a critical point, hence $df=0$ which implies $T_pM$ is inside the kernel of $\langle ., w \rangle$ we conclude that $w$ is orthogonal to $T_pM$
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