Suppose I have a (finite-dimensional) Lie group $(G,\circ)$ with identity element $e\in G$. Then I can always construct a left-invariant metric
$$ g_q\colon T_qG\times T_gG \to \mathbb [0,\infty),\qquad (x,y)\mapsto g_q(x,y) = \langle dL_{q^{-1}}(q)\;x, dL_{q^{-1}}(q)\;y\rangle, $$
where
$$L_q\colon G \to G, \quad p\mapsto L_q(p) = q\circ p$$
is the left-translation (and $dL_q(p)\colon T_pG \to T_{L_q(p)}G = T_{q\circ p}G$ is its derivative) and $\langle\bullet,\bullet\rangle\colon T_eG\times T_eG \to [0,\infty)$ is a scalar product on the Lie algebra $T_eG$, which is a linear space.
From a metric $g_p$ we can construct a distance function $dst$ on $G$, which makes $(G,dst)$ a metric space:
$$ dst\colon G\times G\to [0,\infty),\qquad (q,p)\mapsto dst(q,p) = \inf_{\gamma\in\Gamma(q,p)} L(\gamma),$$
where $\Gamma(q,p)\subseteq G$ is the set of all differentiable curves with $\gamma(0)=q$, $\gamma(1) = p$ and $L(\gamma)$ gives the length of a curve by
$$ L(\gamma) = \int_0^1 \sqrt{g_{\gamma(s)}(\gamma'(s),\gamma'(s))}\;ds. $$
If $g_p$ is left-invariant, then $dst$ is also left-invariant in the sense that
$$\begin{align}dst(q\circ a, q\circ b) = dst(a,b). \tag{9.1}\end{align}$$
I know that not every Lie group admits a bi-invariant metric (for example $SE(3)$ does not, since it is not the direct product of linear and compact Lie groups). Therefore, not every Lie group is a metric space, where the distance is bi-invariant.
I just read "Lie Group Methods" from Iserles, Munthe-Kaas, Nørsett and Zanna and there they state
"[A]ccording to the Birkhoff–Kakutani
theorem (Birkhoff 1936), every Lie group $G$ admits a left-invariant, al-
most right-invariant metric which, in addition to (9.1), obeys
$$ dst(X\circ Z, Y\circ Z) \leq \rho(Z) dst(X, Y),$$
where the function $\rho$ is finite."
(Note that I altered the name of the distance function and used $\circ$ for the Lie group product)
Unfortunately, I don't really understand the paper "A note on topological groups" from Birkhoff (Compositio Mathematica, Volume 3 (1936), p. 427-430) and the Birkhoff-Kakutani seems to be a theorem about whether a toplogical group (or Hausdorff group) is metricizable. I know that a Lie group is a special case of a topological group, but the theorem or the proof do not seem to be concerned with the invariance of the metric.
Can somebody explain to me or point me to a resource, where the existence of a left-invariant and almost right-invariant distance function is discussed? Also, does "$\rho$ is finite" mean that there is a constant $C$ such that $\rho(p)\leq C$ for all $p\in G$?
Best Answer
It's easy, namely due to the fact that every operator in finite dimension has a finite norm.
Indeed, fix a Euclidean structure on $T_1G$. For $g\in G$, the conjugation map $h\mapsto ghg^{-1}$ induces an operator on $T_1G$, with some norm $C_g$ with respect to the Euclidean distance. Then it follows the right translation by $g$ is $C_g$-Lipschitz on $G$.