The group of isometries of Shahshahani metric Edit: 28 January 2023 I just realized that this metric is frequently used in this paper
https://hal.science/hal-01382281/document
Let  $$M=\{(x_1,x_2,\ldots,x_n)\in \mathbb{R}^n\mid x_i>0,\;i=1,2,\ldots,n\}$$
For  $X=(x_1,x_2,\ldots,x_n)\in M$ put $|X|=\sum_{i=1}^n x_i$.
We  consider the  Shahshahani Riemannian metric $g$ on $M$  with  diagonal  tensor  metric $g_{ii}=\frac{|X|}{x_i}$.
What is the  dimension  and  the precise structure  of the  group of  all isometries of  $(M,g)$?
 A: The point is that, if you set $x_i = {u_i}^2$ where $u_i>0$, this becomes a diffeomorphism of $M$ with itself with the property that, in the $u$-coordinates, the Shahshahani metric becomes
$$
g = 4({u_1}^2+\cdots+{u_n}^2)\bigl({\mathrm{d}u_1}^2+\cdots + {\mathrm{d}u_n}^2\bigr).
$$
Clearly, this metric is just the flat metric in the $u$-coordinates on the positive orthant times the squared '$u$-distance' from the origin.  When $n\ge2$, any rotation in the $u$-coordinates would preserve the metric on the entire $\mathbb{R}^n$, but it wouldn't preserve the positive orthant, which is $M$.
When $n=2$, this is a flat metric:  Set $z = u_1 {+} i\,u_2$. then $g = 4 z\bar z\, \mathrm{d}z\circ\mathrm{d}\bar z =\mathrm{d}w\circ\mathrm{d}\bar w$ where $w = z^2 = (u_1 + i\,u_2)^2 = ({u_1}^2{-}{u_2}^2)+ i\,(2u_1u_2)$.  Thus, $w:M\to\mathbb{C}$ isometrically embeds $(M,g)$ as the upper half-plane in $\mathbb{C}$ when $\mathbb{C}$ is given its standard metric.  The global isometries are the translations by a real number in the $w$-coordinate together with reflection in the imaginary axis. Meanwhile, the Lie algebra of Killing fields has dimension $3$ instead of $1$.
However, when $n>2$, the metric is not flat, and the Killing fields are the infinitesimal generators of the obvious $\mathrm{SO}(n)$-action, as can be seen by direct calculation or conversion to 'polar coordinates'.  Meanwhile, globally on $M$, you only have the rotations and reflections (in the $u$-coordinates) that preserve the positive orthant, and this is just the permutations in the $u_i$, which is the permutations in the $x_i$.
