Question

Let $\omega$ be an $n$-form on $S^n$, nowhere vanishing.

Is there a Riemannian metric $g$ on $S^n$, so that its volume form is $\omega$, and $(S^n,g)$ is homogeneous? Is it unique, and if not, what transformations of $S^n$ relates such metrics?

For example, for $n=1$, the unique metric with these properties is $\text{d} u^2$, where $u=\frac{1}{2\pi}\int\omega$.

Answer 1

There is. By Moser's Theorem, there exists a diffeomorphism $f:S^n\to S^n$ such that $f^*\omega = c\omega_{std}$ where $c>0$ and $\omega_{std}$ is the standard volume form homogeneous under $\mathrm{SO}(n{+}1)$. Now use $f$ to transfer the right multiple of the standard metric (under which $S^n$ is homogeneous) back to one for which $\omega$ is the volume form.

About uniqueness: (I forgot to answer this question on the first pass; also I am fixing a few errors in the cases below in this edit.) The answer about uniqueness of homogeneous metrics with a given volume form depends on the dimension $n$. Obviously, when $n=1$, it is unique. It's also unique (up to volume-preserving diffeomorphism) when $n=2$ (and, in fact, for all even $n$). However, it is not unique (up to volume-preserving diffeomorphism) when $n$ is odd.

The reason has to do with this: If a metric $g$ on $S^n$ is homogeneous under some (connected) group $G$ of $g$-isometries, then $S^n = G/H$ where $H\subset G$ is a closed, connected subgroup of $G$ acting effectively on $S^n$. It is known, from the work of Borel, which such pairs $(G,H)$ have the property that $G/H$ is diffeomorphic to $S^n$ for some $n$, so we know all of the possibilities and can work out the isometry groups in each case.

When $n=2k$, the only possibility is $G = \mathrm{SO}(2k{+}1)$ with $H = \mathrm{SO}(2k)$. (While $S^6 = \mathrm{G}_2/\mathrm{SU}(3)$ would appear to be a counterexample, in this case $\mathrm{G}_2$ is not the full isometry group of the invariant metric, $\mathrm{SO}(7)$ is.)

When $n = 4k{+}1>1$, we can have $G = \mathrm{SU}(2k{+}1)$ and $H = \mathrm{SU}(2k)$, and there is a $1$-parameter family of inequivalent metrics homogeneous under $\mathrm{U}(2k{+}1)$ with the same volume form.

When $n=4k{+}3$, we can have $G = \mathrm{Sp}(k{+}1)$ and $H=\mathrm{Sp}(k)$, and there is a $5$- (when $k=0$) or $6$- (when $k>0$) parameter family of $G$-invariant on $S^n$ with a given $G$-invariant volume form. (Some of these are isometric though; you can get rid of $3$ of the parameters that way.)

Finally, there is the exceptional case of $S^{15} = \mathrm{Spin}(9)/\mathrm{Spin}(7)$, which has a 1-parameter family of inequivalent metrics with the same volume form. (Even though we also have $S^{15}=\mathrm{Sp}(4)/\mathrm{Sp}(3)$, these homogeneous metrics are not comparable, except at the constant curvature metric, which belongs to both families.