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My book is Connections, Curvature, and Characteristic Classes by Loring W. Tu (I'll call this Volume 3), a sequel to both Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott (Volume 2) and An Introduction to Manifolds by Loring W. Tu (Volume 1).

I refer to the last paragraph of Section 5.1.

Some context: Volume 1 Section 8.9, Volume 1 Section 9.2, Volume 1 Lemma 9.7, Volume 1 Theorem 9.9 (Regular Level Set Theorem)

I think the last paragraph says that

  • A. Surfaces (regular 2-submanifolds of $\mathbb R^3$), locally, are zero sets of "coordinate functions" and then

  • B. Proposition 5.1 applies, locally, to surfaces (to say $N$'s exist locally on surfaces) assuming (C) below.

  • C. The zero sets in (A) are regular zero sets.

Here, I attempt to prove $A$ and $C$. Question: Are these proofs correct? Please verify.

To prove (A):

  • For the term "coordinate function", I assume for a chart $(W,\varphi)$ about a point $b$ in an n-manifold $B$, $W$ is the coordinate neighborhood and $\varphi$ is the coordinate map so that by "coordinate function", Tu refers to a component of a coordinate map, so the coordinate functions of the coordinate map $\varphi=(x^1, ..., x^n)$ are $x^1, x^2, ..., x^{n-1}$ and $x^n$.

We must show that for each $p \in M$, there exists a neighborhood $U$ of $p$ in $M$ such that $U=\{a=0\}$ for some $a$ which is a coordinate function of $\varphi=(a,b,c)$ where $\varphi$ is a coordinate map in a coordinate chart $(W,\varphi)$ about $p$ in $\mathbb R^3$.

  1. For each $p \in M$, there exists a coordinate chart $(V,x^1,x^2,x^3)$ about $p$ in $\mathbb R^3$ such that $V \cap M = \{x^i=0\}$ for exactly 1 index $i = 1,2,3$.

  2. $V \cap M$ is open in $M$.

  3. Therefore, from (1), we have that $V \cap M$, with $a=x^i$, satisfies all the properties required of $U$ except for the "neighborhood". From (2), we can choose $U = V \cap M$.

To prove (C):

  1. We have for $x^i: V \to \mathbb R$ that $0$ is a regular point of $x^i$ if and only if $(x^i)^{-1}(0) = \{x^i = 0\}$ is a regular zero set.

  2. By Volume 1 Proposition 8.23, we have for $x^i: V \to \mathbb R$ that $0$ is a regular point of $x^i$ if and only if there exists a chart $(A,y^1,y^2,y^3)$ about $p$ in $V$ such that $\frac{\partial x^i}{\partial y^j}|_p \ne 0$ for some index $j \in \{1,2,3\}$.

  3. By Volume 1 Example 5.12, if $P$ is an open subset of an $n$-manifold $Q$ and if there exists a chart $(P,\gamma)$ in the maximal atlas of $Q$, then $(P,\gamma)$ is also in the maximal atlas of $P$. (I believe this is indeed precisely the same $(P,\gamma)$ and not $(P \cap \text{something},\gamma|_{P \cap \text{something}})$.)

  4. By (6), $(V,x^1,x^2,x^3)$ is in not only the maximal atlas of $\mathbb R^3$ but also the maximal atlas of $V$.

  5. By (7), we may choose for (5) that $(A,y^1,y^2,y^3) = (V,x^1,x^2,x^3)$.

  6. By (8), we have $\frac{\partial x^i}{\partial y^j}|_p$ is 0 for $j \ne i$ and is 1 for $j = i$. (I believe the usual notation is something like $\frac{\partial x^i}{\partial y^j}|_p = \delta^i_{j}$.)

  7. By (5) and (9), $0$ is a regular point of $x^i$.

  8. Therefore, by (10) and (4), $(x^i)^{-1}(0)$ is a regular zero set.

Related:

Is every embedded submanifold globally a level set?

Every submanifold of $\mathbb R^n$ is locally a level set

(My question) Regular submanifold is locally a level set?

  • Did you show that given a smooth manifold $X$ of dimension $n$ and some smooth function $f : X \to \Bbb{R}$, if for every $p \in f^{-1}(0)$ one of the partial derivative $D_v f(p) \ne 0$ then any connected component of $f^{-1}(0)$ is locally a smooth regular submanifold of dimension $n-1$ ? Conversely if $H$ is a smooth hypersurface, when is it (locally) the vanishing set of such a $f$ ? What about the $n-m$ dimensional submanifolds vanishing sets of smooth functions $X \to \Bbb{R}^m$ – reuns Aug 22 '19 at 03:26
  • @reuns My question is a proof verification. Are my proofs correct or not? –  Aug 23 '19 at 05:01

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