Je bent je ingevulde velden bij deze pagina aan het verwijderen. Ben je zeker dat je dit wilt doen?
You are erasing your filled-in fields on this page. Are you sure that is what you want?
Nieuwe Versie BeschikbaarNew Version Available
Er is een update van deze pagina. Als je update naar de meest recente versie, verlies je mogelijk je huidige antwoorden voor deze pagina. Hoe wil je verdergaan ?
There is an updated version of this page. If you update to the most recent version, then your current progress on this page will be erased. Regardless, your record of completion will remain. How would you like to proceed?
In each exercise, you can use the previous exercises/items.
Let \(M\) be a smooth manifold and \(V \in \mathfrak {X}(M)\) a vector field. Assume there is \(\varepsilon > 0\), a collection \(\{ W_i \} _{i \in I}\) of
open sets of \(M\), and a collection of maps \( \{ \Phi _i : W_i \times (-\varepsilon , \varepsilon ) \to M \} _{i \in I } \) such that \(\Phi _i\) is a local flow of \(V\) and
\[ M \subset \bigcup _{i \in I} W_i . \]
Show that
there is a flow \(\Phi : M \times \mathbb {R} \to M\) of \(V\) (defined everywhere for all time).
Let \(M\) be a smooth manifold and \(V \in \mathfrak {X}(M)\) a vector field. The support supp\((V)\) of \(V\) is defined as
the topological closure of the set
\[ \{ x \in M \, \vert \, V( x) \neq 0 \} . \]
Show that if supp\((V)\) is compact, then there is a flow \(\Phi : M \times \mathbb {R} \to M\)
of \(V\) (defined everywhere for all time).
Let \(M\) be a connected smooth manifold. Show that for any points \(p , q \in M\) there is a
diffeomorphism \(f : M \to M\) with \(f(p) = q\).
Let \(G\) be a smooth manifold and a group. We say \(G\) is a Lie group if both the
multiplication \(G \times G \to G\) and the inverse map \(G \to G\) are smooth.
Let \(G\) be a Lie group.
Show that for each \(g \in G\), the map \(L_g : G \to G\) given by \(L_g(h) : = gh\) is a diffeomorphism.
Fix \(v \in T_eG\). For each \(g \in G\), define \(V(g) \in T_gG\) as
\[ V(g) : = (L_g)_{\ast }(v).\]
Show that \(V\) is a smooth vector field and that \(V\)
is \(L_h\)-related to itself for each \(h \in G\).
Take \(V \in \mathfrak {X}(G)\) as in the previous item. Show that there is a flow \(\Phi : G \times \mathbb {R} \to G\) of \(V\) (defined
everywhere for all time).
With \(V\in \mathfrak {X}(G)\) as in the previous exercises, take the flow line \(\gamma : \mathbb {R} \to G\) with \(\gamma (0) = e\). Show that \(\gamma \) is
a group morphism from \(\mathbb {R} \) to \(G\). We write
\[ \exp (t v) : = \gamma (t) . \]
Show that \(GL(n;\mathbb {R}) : = \{ A \in M_n (\mathbb {R}) \, \vert \, \det (A) \neq 0 \} \) is a Lie group. Then show that for any \(A \in M_n(\mathbb {R}) = T_{Id} GL(n; \mathbb {R})\) one has
\[ \exp ( A ) = e ^A : = \sum _{j = 0 }^{\infty } \frac {A^k}{k!} . \]
Smooth manifolds