Exercise 3, § 3

Let $n$ be a natural number. We make the topological group SO($n+ 1,\mathbf{R}$) act on the sphere $\mathbf{S}_n$ of $\mathbf{R}^{n+1}$.

a) Exercise prove that the stabilizer of the vector $\mathbf{e}_{n+1}$ is identified with the topological group SO($n,\mathbf{R}$).

b) Let $f:\mathbf{I}\rightarrow \mathbf{S}_n$ be a continuous mapping. We suppose that $f(t)=\not$ $(0, . . . ,0,1)$ for every $t\in \mathbf{I}$. Exercise prove that there exists a continuous mapping $g:\mathbf{I}\rightarrow$ SO($n+ 1,\mathbf{R}$) such that $f(t) =g(t)\cdot \mathbf{e}_{n+1}$ for every $t\in \mathbf{I}$.

c) If $n\geqslant 3$, exercise prove that the injection of SO($n,\mathbf{R}$) into SO($n+1,\mathbf{R}$) induces an isomorphism between Poincaré groups.