## 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

$\left\{ \gamma'(0), \gamma \in \Gamma_0 \right\}$

is a linear space isomorphic to $TM_x$.

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