Exercise 1. Show that the sphere
is a -dimensional smooth manifold.
Exercise 2. We consider the following two subsets of the plane
Are and smooth manifolds ? Of course, justify your answer with a proof.
Exercise 3. Let be a n-dimensional smooth manifold. Let . A smooth curve on is a smooth map . We denote by the set of smooth curves on such that . Show that the set
is a linear space isomorphic to .