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Submanifolds. Examples given by embeddings and level sets.
A reference for this material is Chapter 5 of John M. Lee. Introduction to smooth
manifolds. Second edition. Grad. Texts in Math., 218. Springer, New York, 2013.
xvi+708pp. ISBN: 978-1-4419-9981-8.
Submanifold Let \(M\) be an \(n\)-dimensional smooth manifold, \(k \in \mathbb {N}\), and \(S \subset M\). We say that \(S\) is a
\(k\)-dimensional submanifold of \(M\) if for any \(p \in S\) there is a chart \((W, \varphi )\) of \(M\) such that \(p \in W\) and
\begin{equation}\label {eq:submanifold} \varphi (S \cap W ) = \varphi (W) \cap ( \mathbb {R}^k \times \{ 0^{n-k} \} ), \end{equation}
where \(0^{n-k} \in \mathbb {R}^{n-k}\)
denotes the zero vector. The difference \(n-k\) is called the codimension of \(S\) in \(M\).
Submanifold is smooth manifold Let \(M\) be an \(n\)-dimensional manifold, \(S \subset M\) a \(k\)-dimensional
submanifold, and \(\mathcal {A}\) the set of charts \((W, \varphi )\) of \(M\) satisfying (eq:submanifold). Then the set
Local graphs Let \(S \subset \mathbb {R}^{k + m }\) be a set that is locally a \(k\)-dimensional graph (see Example ex:local-graph in
the Examples section of part I). Then \(S\) is a \(k\)-dimensional submanifold of
\(\mathbb {R}^{k+m}\).
Solution:
By hypothesis, for each \(p \in S\), there is a linear isomorphism \(L : \mathbb {R}^{k+m} \to \mathbb {R}^{k+m}\), open sets \(U \subset \mathbb {R}^k \), \(V \subset \mathbb {R}^m\), and a smooth
function \(f \in C^{\infty }(U ; \mathbb {R}^m)\) such that \(L(p) \in U \times V\) and
\begin{equation}\label {eq:local-graph} L (S ) \cap (U \times V) = \{ (x, f(x) ) \in \mathbb {R}^k \times \mathbb {R}^m \, \vert \, x \in U \} . \end{equation}
Let \(W : = L ^{-1} ( U \times V ) \subset \mathbb {R}^{k + m } \) and \(\varphi : W \to \mathbb {R}^{k + m}\) given by
\[ \varphi ( L^{-1} (x, y ) ) = (x , y - f (x ) ). \]
On one hand, if \(p \in S \cap W \), then \(L(p) \in L(S ) \cap (U \times V )\). By (eq:local-graph), we then
have \(L (p ) = (x,f(x)) \) for some \(x \in U\). Hence
Images of embeddings Let \(S\) be a smooth \(k\)-dimensional manifold, \(M\) a smooth
\(n\)-dimensional manifold, and \(f : S \to M\) be a smooth embedding. Then \(f(S) \subset M\) is a \(k\)-dimensional
submanifold.
Proof in the case \(M = \mathbb {R}^n\) Fix \(p \in S\). By hypothesis, the vector subspace
\[ V : = d_p f (T_p S) \leq \mathbb {R}^n \]
is \(k\)-dimensional. Let \(\Vec {v}_1 , \ldots , \Vec {v}_{n-k} \in \mathbb {R}^n \) be a
basis of a complement of \(V\). Let \((U,\psi ) \) be a chart of \(S\) around \(p\) with \(\psi (p) = 0 \), and define \( h : \psi (U ) \times \mathbb {R}^{n-k} \to \mathbb {R}^n \) by
\[ \frac {\partial h }{\partial x_1} (0) , \ldots , \frac {\partial h }{ \partial x_n } (0) \]
span \(V\), and the partial derivatives
\[ \frac {\partial h }{\partial a_1} (0) , \ldots , \frac {\partial h }{ \partial a_{n-k} } (0) \]
span a complement of \(V\).
Therefore, by the inverse function theorem, \(h\) is a local diffeomorphism around \(0\). This
means there are \(W_0 \subset \mathbb {R}^k\), \(W_1 \subset \mathbb {R}^{n-k}\) open sets containing \(0\) such that \(h\) restricted to \(W_0 \times W_1\) is a diffeomorphism.
Since \(f\) is a topological embedding, after shrinking both \(W_0\) and \(W_1\), we can guarantee
\begin{equation}\label {eq:submanifold-embedding} f (S) \cap h (W_0 \times W_1) = h(W_0 \times \{ 0 \} ) . \end{equation}
Then
we can define \(W : = h (W_ 0 \times W_1) \), and \(\varphi : W \to \mathbb {R}^n\) by \( \varphi : = h^{-1} \). By (eq:submanifold-embedding), the chart \((W , \varphi )\) satisfies the condition in the
definition of submanifold. Since \(p \in S \) was arbitrary, \(f(S) \subset \mathbb {R}^n\) is a \(k\)-dimensional submanifold.
Proof of the general case Exercise.
Level sets of submersions Let \(M^m\), \(N^n\) be smooth manifolds, \(F : M \to N\) a submersion, and \(q \in N\). Then \(F^{-1} (q) \subset M\) is
an \((m-n)\)-dimensional submanifold.
Solution:
Let \(p \in F^{-1} (q)\), \((V, \psi _2 )\) a chart of \(N\) with \(\psi _2 (q) = 0\), and \((\Omega , \psi _1 )\) a chart of \(M \) around \(p\) with \(F(\Omega ) \subset V\). By hypothesis, the
linear map
is surjective. By the implicit function theorem, \(\psi _1 ( F^{-1}(q) \cap \Omega )\) is locally a graph
in \(\psi _1 (\Omega )\). By the first example, \(F^{-1}(q) \cap \Omega \) is an \((m-n)\)-dimensional submanifold of \(M\). Finally, being a
submanifold is a local property, so the result follows.
Preimage of regular point Let \(M^m\), \(N^n\) be smooth manifolds, \(F : M \to N\) a smooth map, and \(q \in N\) such that
\(d_pF : T_p M \to T_{q}N\) is surjective for all \(p \in F^{-1}(q)\). Then \(F^{-1} (q) \subset M\) is an \((m-n)\)-dimensional submanifold.
Solution:
Proof is identical to the one of the previous example.
Level sets of constant rank maps Let \(r \in \mathbb {N}\) and \(F : M^m \to N^n\) a smooth map with \(\text {rank} ( d_p F ) = r \) for all \(p\in M\). Then for all \(q \in N \),
the preimage \(F^{-1}(q) \subset M \) is a smooth \((m-r)\)-dimensional submanifold.
Solution:
Can be found in Sections 4 and 5 of the book. Specifically, Theorem 5.12.
Smooth manifolds