Proposition 12.6.1.88C1708A920C The map
\[ \Phi :\mathbb R\times \mathscr S\longrightarrow \mathscr C, \qquad (r,q)\longmapsto \mathrm{e}^ r q \]
is an isometry from the product metric $\mathrm{d}r^2+g_{\mathscr S}$ onto $g$. In particular, if $d_ g$ and $d_{\mathscr S}$ are the intrinsic distances, then
\[ d_ g(\mathrm{e}^{r_0}q_0,\mathrm{e}^{r_1}q_1)^2 =|r_1-r_0|^2+d_{\mathscr S}(q_0,q_1)^2. \]
The metric on $\mathscr C$ is complete if and only if the induced metric on $\mathscr S$ is complete. Every path for which $\lambda $ tends to zero or to infinity has infinite length.
Proof.
Homogeneity gives $\lambda (s\theta )=s\lambda (\theta )$ for $s>0$. Hence $r=\log \lambda (\theta )$ and $q=\theta /\lambda (\theta )$ define the smooth inverse of $\Phi $. The level set is smooth because the Euler identity gives $D\lambda _\theta (\theta )=\lambda (\theta )>0$.
Differentiating the Euler identity $D\log \lambda _\theta (\theta )=1$ in a direction $\chi \in V$ gives
\[ g_\theta (\theta ,\chi )=D\log \lambda _\theta (\chi ), \qquad g_\theta (\theta ,\theta )=1. \]
For directions $\chi ,\eta \in V$, differentiating homogeneity twice also gives
\[ g_{s\theta }(s\chi ,s\eta )=g_\theta (\chi ,\eta ). \]
Let $q\in \mathscr S$, let $\xi ,\eta \in T_ q\mathscr S$, and let $a,b,r$ be real numbers. Since $\xi $ and $\eta $ are tangent to a level set of $\lambda $, the first of these identities shows that both are $g_ q$-orthogonal to the radial vector $q$. Using also the second identity and the scaling rule, we obtain
\[ g_{\mathrm{e}^ rq}\bigl (\mathrm{e}^ r(aq+\xi ),\mathrm{e}^ r(bq+\eta )\bigr ) =ab+g_ q(\xi ,\eta ), \]
which proves the metric identity.
It remains to prove the distance formula and its consequences. A path $(r(t),q(t))$ in the product has length $\int \sqrt{|r'|^2+|q'|_{g_{\mathscr S}}^2}\, \mathrm{d}t$. The triangle inequality in $\mathbb R^2$ (applied to the integral of the vector $(|r'|,|q'|_{g_{\mathscr S}})$) bounds this below by the square root of the sum of the squared endpoint difference in $r$ and the squared length of the projected path in $\mathscr S$. For the opposite bound, choose a path in $\mathscr S$ of length within $\epsilon $ of the distance of its endpoints, parametrize it with constant speed, and let $r$ vary linearly; then let $\epsilon $ decrease to zero. This proves the distance formula. It identifies Cauchy sequences in $\mathscr C$ with pairs of Cauchy sequences in the two factors; since $\mathbb R$ is complete, the completeness assertion follows. Finally, the length of any path is at least the total change of $\log \lambda $ along it, and this change is infinite when $\lambda $ tends to zero or to infinity.
$\square$
Pome
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