**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 $(M,g)$?



The group of isometries of Shahshahani metric
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 $x\_i$.


