Lagrange multipliers on manifolds in Lee's book

Question

Here is Problem 11-11 on page 301 of John Lee’s book:

Let $ M $ be a smooth manifold, and $ C \subset M $ be an embedded sub-manifold. Let $ f \in {C^{\infty}}(M) $, and suppose $ p \in C $ is a point at which $ f $ attains a local maximum or minimum value among points in $ C $. Given a smooth local defining function $ \Phi: U \to \mathbb{R}^{k} $ for $ C $ on a neighborhood $ U $ of $ p $ in $ M $, there are real numbers $ \lambda_{1},\ldots,\lambda_{k} $ (called Lagrange multipliers) such that $$ \mathrm{d} f_{p} = \sum_{i = 1}^{k} \lambda_{i} \cdot \mathrm{d} \Phi^{i}|_{p}. $$

I got confused when I was trying to solve it. Here are my questions:

(1) He didn’t say anything about the dimension of $ M $ and $ C $, nor did he put corrections here. Is it necessary to assume that $ \operatorname{dim}(M) = n > k $ and $ \operatorname{dim}(C) = n - k $, or do these results implicitly follow from the conditions of this problem?

(2) Why do we need the condition that $ C $ is an embedded sub-manifold? Assume $ \operatorname{dim}(C) = n - k $; then Theorem 5.8 on page 102 tells us that $ C $ satisfies the local $ k $-slice condition (this is the only theorem I can think of that is related to this condition), but what good can this condition do for us?

(3) I think I need to apply the Lagrange Multiplier Theorem (see page 113) in multi-variable calculus, but we need to make sure that the rank of the Jacobian matrix of $ \Phi $ is of rank $ k $ at the point $ p $. However, there aren’t any extra conditions on $ \Phi $.

You can either answer my questions separately or show me a detailed proof of it. Thank you in advance!

Answer 1

Here are a few comments that might be helpful.

(1) He didn't say anything about the dimension of $M$ and $C$, nor did he put corrections [here][1]. Is it necessary to assume that $dim > M=n>k$ and that $dim C=n-k$ or these results implicitly follow from the conditions in this problem?

The definition of a local defining function (page 107) specifies that $C\cap U$ is a regular level set of $\Phi$, which implies that $d\Phi$ has rank $k$ everywhere on $C\cap U$, and therefore $C$ has codimension $k$ in $M$.

(2) Why do we need the condition that $C$ is an embedded submanifold?

Embedded submanifolds are the only ones that admit local defining functions in a neighborhood of each point.

(3) I think I need to apply the Lagrange multipliers theorem in multi-variable calculus. But we need to make sure that the rank of the Jacobian matrix of $\Phi$ is of rank $k$ at the point $p$. However, there isn't any extra conditions on $\Phi$.

The point of this problem is to prove the Lagrange multiplier theorem, albeit in a more general setting than the one usually introduced in advanced calculus courses. The fact that the Jacobian of $\Phi$ in coordinates has maximal rank is an immediate consequence of the definition of a local defining function.