Exercise 11, § 9

For any integer $n\geq 1$, define a map $\varphi_n$ from $(0,\frac{1}{2}(\times \mathbf{T}^2$ to $\mathbf{R}^3$ by putting $\varphi_n(r;\alpha , \beta ) = ((n+r$ cos $2\pi \beta )$ cos $2\pi \alpha ,(n+r$ cos $2\pi \beta )$ sin $2\pi \alpha , r$ sin $2\pi \beta )$. Put

a) Show that the restriction of $\varphi_n$ to $0,\frac{1}{2}\times \mathbf{T}^2$ is an isomorphism (of class $C^{\omega}$) from $0,\frac{1}{2}\times \mathbf{T}^2$ to $T_n$ -- $S_n$.

b) Let $f_n: T_n\rightarrow T_n$ be the map that coincides with the identity on $S_n$ and is such that

for $r >0$. Show that $f_n$ induces an automorphism of $T_n$ -- $S_n$.

c) Show that there exists a real-analytic manifold structure on $\mathbf{R}^3$ such that the maps $f_n: T_n\rightarrow \mathbf{R}^3$ and the canonical injection $\mathbf{R}^3$ -- $\bigcup_nS_n\rightarrow \mathbf{R}^3$ are analytic. Denote by X the real-analytic manifold defined in this way.

d) For $\vartheta \in \mathbf{T}$, denote by $R_{\vartheta}$ the rotation

$(x, y, z) \rightarrow (x$ cos $2\pi \vartheta -y$ sin $2\pi \vartheta , x$ sin $2\pi \vartheta +y$ cos $2\pi \vartheta , z)$

of $\mathbf{R}^3$. Prove that putting, for $\theta \in \mathbf{T}$ and $u\in X$,

$\vartheta .u= R_{\vartheta}(u)$ if $u\in X$ -- $\bigcup_nS_n$;

$\vartheta .u= R_{n\vartheta}(u)$ if $u\in S_n$ for some integer $n\geq 1$

defines an analytic law of operation of $\mathbf{T}$ on X.

e) Show that the set of orbit types of X (for the operation of $\mathbf{T}$ defined in $d$)$)$ is infinite. f) Show that there is no embedding, compatible with the operation of $\mathbf{T}$, of X in a finite dimensional vector space on which $\mathbf{T}$ operates linearly (use Exerc. 9).