MA5311: Homework 1, due Wednesday 1/25

Exercise 1. Show that the sphere

$\mathbf{S}^n=\left\{ (x_1,\cdots,x_{n+1}) \in \mathbf{R}^{n+1}, x_1^2+\cdots+x_{n+1}^2=1 \right\}$

is a $n$ -dimensional smooth manifold.

Exercise 2. We consider the following two subsets of the plane

$A= \{ ( x , \sin (1/x)), x \neq 0 \}$

and

$B= \{ ( x , | x| ), x \in \mathbf{R} \}$ .

Are $A$ and $B$ smooth manifolds ? Of course, justify your answer with a proof.

Exercise 3. Let $M \subset\mathbf{R}^{k}$ be a n-dimensional smooth manifold. Let $x \in M$ . A smooth curve on $M$ is a smooth map $\gamma: \mathbf{R} \to M$ . We denote by $\Gamma_0$ the set of smooth curves on $M$ such that $\gamma (0)=x$ . Show that the set