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\(\mathbb {R}^n\) is a smooth manifold in a canonical way.
Solution:
\(\mathbb {R}^n\) 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.
Solution:
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 \(f : U \to \mathbb {R}^m\) 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.
Solution:
Let \(L \), \(U\), \(V\), and \(f\) as above. Define
\[ \varphi : M \cap L^{-1} (U \times V) \to U \]
as
\[ \varphi : = \pi \circ L \]
where \(\pi : \mathbb {R}^n \times \mathbb {R}^m \to \mathbb {R}^n\) denotes the projection onto the first \(n\)
coordinates. Notice that
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
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.
Solution:
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.
Solution:
For each chart \((U, \varphi )\) of \(M\) and each chart \((V, \psi ) \) of \(N\), we consider the map
\[ \varphi \otimes \psi : U \times V \to \varphi (U) \times \psi (V) \subset \mathbb {R}^{n+m} \]
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.
Solution:
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
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 \(n(n-1)/2\).
Solution:
Homework.
Connected sum Let \(M\), \(N\) be connected smooth \(n\)-dimensional manifolds, \((U, \varphi )\) a chart of \(M\), and \((V, \psi )\)
a chart of \(N\) such that
where \(\sim \) identifies \(\varphi ^{-1} (x)\) with \(\psi ^{-1} \left ( \frac {3 - \Vert x \Vert }{\Vert x \Vert } \, x \right )\) for \(x \in B_2 (0) \backslash \overline {B}_1(0)\).
Let \(\mathcal {A}_M\) denote the set of charts \((W, \alpha )\) of \(M\) with \(W \cap \varphi ^{-1} (\overline {B}_1(0)) = \emptyset \) and \(\mathcal {A}_N\) the set of charts \((Z, \beta )\) of \(N\) with \(Z \cap \psi ^{-1} (\overline { B}_1(0)) = \emptyset \).
Finally, define \(U ' : = \varphi ^{-1} (B_2 (0) \backslash \overline {B}_1(0))\) and \(V ' : = \psi ^{-1} (B_2(0) \backslash \overline {B}_1(0))\). Then the set
Smooth manifolds