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 \}


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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