Due April 10.

Let \(M\) be an \(n\)-dimensional topological manifold. Show that being compatible is an equivalence relation in the set of charts of \(M\).
Let \(M\) be an \(n\)-dimensional topological manifold and \(\mathcal {A}\) a smooth atlas. Show that there is a unique smooth structure \(\mathcal {S}\) containing \(\mathcal {A}\). Moreover, show that \(\mathcal {S}\) consists precisely of the charts compatible with all charts in \(\mathcal {A}\).
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.
Consider a chart \((U, \varphi )\) of \(M\) with \(U \subset M \) an evenly covered neighborhood. For each section \( \sigma : U \to \tilde {M} \), consider the chart \((\sigma (U) , \varphi \circ \pi )\). Show that the set of charts of \(\tilde {M}\) obtained this way is a smooth atlas. Recall that a section is a continuous map \(\sigma : U \to \tilde {M}\) with \(\pi \circ \sigma = \text {Id}_U\).
Let \(M_n (\mathbb {R})\) denote the space of \(n\times n\) real matrices and identify it with \(\mathbb {R}^{n^2}\). Show that
\[ SL (n ; \mathbb {R} ) : = \{ A \in M_n (\mathbb {R}) \, \vert \, \det (A) = 1 \} \]
is a smooth manifold of dimension \(n^2 -1 \).
Given \(A \in M_n(\mathbb {R})\), we denote by \(A^T \) its transpose. Show that
\[ O (n ) : = \{ A \in M_n (\mathbb {R}) \, \vert \, A A^{T} = 1 \} \]
is a smooth manifold of dimension \(n(n-1)/2\).
Identify \(M_n (\mathbb {R}) = \mathbb {R}^{n^2}\) with \((\mathbb {R}^n)^n\), the set of \(n\)-tuples of elements of \(\mathbb {R}^n\), and \(\mathbb {R}^{n(n+1)/2}\) with the set of upper-triangular \(n \times n\) matrices. Define
\[ F : (\mathbb {R}^n)^n \to \mathbb {R}^{n(n+1)/2} \]
by
\[ F _{ij}( \Vec {v}_1, \ldots , \Vec {v}_n ) = \Vec {v}_i \cdot \Vec {v}_j - \delta _{ij} \]
for \(1\leq i \leq j\leq n\), where \(\delta _{ij}\) denotes the Kronecker delta. Show that
\[ F^{-1} (0) = O (n) \]
and that for all \( p \in O (n) \) the differential \(d_p F : (\mathbb {R}^n)^n \to \mathbb {R}^{n(n+1)/2} \) is surjective.