Examples

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Inverse images of regular values, products, covering spaces, orthogonal group

$\mathbb {R}^n$ is a smooth manifold in a canonical way.

$\mathbb {R}^n$ is a metric space, hence Hausdorff. The set of balls with rational radius and center with rational coordinates forms a countable sub-basis of the topology of $\mathbb {R}^n$, so it is second countable.

The identity map $\text {Id}_{\mathbb {R}^n} : \mathbb {R}^n \to \mathbb {R}^n$ is a chart. Since the domain is everything, and any chart is compatible with itself, the set containing this chart is a smooth atlas. More specifically, $\mathcal {A} : = \{ (\mathbb {R}^n, \text {Id}_{\mathbb {R}^n} ) \}$ is a smooth atlas of the topological manifold $\mathbb {R}^n$.

Therefore, there is a unique smooth structure containing this chart. Note that for each open set $U \subset \mathbb {R}^n$, the inclusion $U \hookrightarrow \mathbb {R}^n$ belongs to this smooth structure.

Open sets Let $M$ be an $n$-dimensional smooth manifold and $U \subset M$ an open set. Then $U$ is an $n$-dimensional smooth manifold.

Proof sketch Let $\mathcal {S} = \{ (U_i, \varphi _i ) \} _{i \in I }$ be the smooth structure of $M$. Then $\mathcal {S}_U : = \{ ( U \cap U_i , \varphi _i \vert _{U \cap U_i} ) \} _{i \in I }$ is a smooth atlas on $U$ (it is actually a smooth structure).

Locally a graph Let $M\subset \mathbb {R}^{n+m}$ be a set that is locally the graph of a smooth function. That is, for each $p \in M$, there is a linear isomorphism $L : \mathbb {R}^{n+m} \to \mathbb {R}^{n+m}$, open sets $U \subset \mathbb {R}^n $, $V \subset \mathbb {R}^m$, and a smooth function $f : U \to \mathbb {R}^n$ such that $L(p) \in U\times V$ and

\[ L (M) \cap ( U \times V ) = \{ (x,f(x)) \in \mathbb {R}^n \times \mathbb {R}^m \, \vert \, x \in U \} . \]

Then $M$ is a smooth manifold.

Let $L : \mathbb {R}$, $U$, $V$, and $f$ as above. Define

\[ \varphi : M \cap L^{-1} (U \times V) \to U \]

\[ \varphi : = \pi \circ L \]

where $\pi : \mathbb {R}^n \times \mathbb {R}^m \to \mathbb {R}^n$ denotes the projection onto the first $n$ coordinates. Notice that

\begin{align*} \varphi (L^{-1} (x, f(x))) & = \pi \circ L \circ L^{-1} (x, f(x)) \\ & = \pi (x, f(x)) \\ & = x . \end{align*}

Hence the inverse $\varphi ^{-1} : U \to M \cap L^{-1} (U \times V)$ is the continuous function

\[ \varphi ^{-1} (x) = L ^{-1} (x, f(x)) , \]

showing that $\varphi $ is a chart. We claim that the set of all charts obtained this way form an atlas. For that purpose, take a linear isomorphism $ \hat {L} : \mathbb {R}^{n+m} \to \mathbb {R}^{n+m}$, open sets $\hat {U} \subset \mathbb {R}^n $, $\hat {V} \subset \mathbb {R}^m$, and a smooth function $\hat {f} : \hat {U} \to \mathbb {R}^n$ such that

\[ \hat {L} (M) \cap ( \hat {U} \times \hat { V} ) = \{ (x,\hat {f}(x)) \in \mathbb {R}^n \times \mathbb {R}^m \, \vert \, x \in \hat {U} \} . \]

Construct with them the chart

\[ \psi : M \times \hat {L}^{-1} (\hat {U} \times \hat {V} ) \to \hat {U} \]

given by

\[ \psi : = \pi \circ \hat {L} . \]

The change of coordinates is then given by

\[ \psi \circ \varphi ^{-1} (x) = ( \pi \circ \hat {L} ) \, ( L ^{-1} ( x, f( x ) ) ) , \]

which is the composition of the smooth functions

\[ x \mapsto ( x, f(x) ) \mapsto L ^{-1} (x, f(x)) \mapsto \hat {L} ( L ^{-1} (x,f(x)) ) \mapsto \pi ( \hat {L} ( L^{-1} (x, f(x)) ) ). \]

Therefore the charts $\varphi $ and $\psi $ are compatible, proving our claim.

Level sets Let $\Omega \subset \mathbb {R}^{n+m} $ be an open set and $F \in C^{\infty } ( \Omega ; \mathbb {R}^m )$. If for all $p \in F^{-1} (0)$, the differential $d_pF : \mathbb {R}^{n+m} \to \mathbb {R}^m$ is surjective, then the level set $F^{-1}(0) \subset \mathbb {R}^{n+m}$ is a smooth manifold.

By the Implicit Function Theorem, this is covered by the above example

Products Let $M$ and $N$ be smooth manifolds of dimension $m$ and $n$, respectively. Then $M \times N$ is a smooth manifold of dimension $m + n$ in a canonical way.

For each chart $(U, \varphi )$ of $M$ and each chart $(V, \psi ) $ of $N$, we consider the map

\[ \varphi \otimes \psi : U \times V \to \varphi (U) \times \psi (V) \subset \mathbb {R}^{n+m} \]

given by

\[ (\varphi \otimes \psi ) (x,y) : = (\varphi (x), \psi (y)). \]

Since products of homeomorphisms are homeomorphisms, $\varphi \otimes \psi $ is a chart.

Now consider $(U', \varphi ' )$ another chart of $M$ and $(V' , \psi ' ) $ another chart of $N$. With those, we build the chart

\[ \varphi ' \otimes \psi ' : U ' \times V' \to \varphi ' (U ' ) \times \psi ' (V ' ) . \]

Notice that

\[ (U \times V) \cap (U ' \times V' ) = (U \cap U') \times (V \times V') . \]

Then the change of coordinates between $\varphi \otimes \psi $ and $\varphi ' \otimes \psi ' $ is the function

\[ (\varphi ' \otimes \psi ') \circ (\varphi \otimes \psi ) ^{-1} : ( \varphi \otimes \psi ) ( ( U \cap U ' ) \times (V \cap V') ) \to ( \varphi ' \otimes \psi ' ) ( ( U \cap U ' ) \times (V \cap V') ) \]

given by

\begin{align*} (\varphi ' \otimes \psi ' ) \circ (\varphi \otimes \psi ) ^{-1} (x,y) & = (\varphi ' \otimes \psi ' ) (\varphi ^{-1} (x) , \psi ^{-1} y) \\ & = ( \varphi ' \circ \varphi ^{-1} (x) , \psi ' \circ \psi ^{-1} (y) ) . \end{align*}

Since the maps $\varphi ' \circ \varphi ^{-1}$ and $\psi ' \circ \psi ^{-1}$ are smooth, the change of coordinates is smooth.

Covering spaces Let $M$ be a smooth manifold and $\pi : \tilde {M} \to M$ a covering map with $\tilde {M}$ second countable. Show that $\tilde {M}$ admits a smooth structure for which $\pi $ is smooth.

Homework.

Special linear group Let $M_n (\mathbb {R})$ denote the space of $n\times n$ real matrices and identify it with $\mathbb {R}^{n^2}$. Then

\[ SL (n ; \mathbb {R}) : = \{ A \in M_n (\mathbb {R}) \, \vert \, \det A = 1 \} \]

is a smooth manifold of dimension $n^2-1$.

Homework.

Orthogonal group Given $A \in M_n(\mathbb {R})$, we denote by $A^T $ its transpose. Then

\[ O (n ) : = \{ A \in M_n (\mathbb {R}) \, \vert \, A A^{T} = \text {Id} \, \} \]

is a smooth manifold of dimension $n(n-1)/2$.

Homework.