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I've been thinking about $SU(n)$-invariant metrics on the odd-dimensional spheres $S^{2n-1} \simeq SU(n)/SU(n-1)$. For $S^1$, all such metrics are in correspondence with the positive reals. For $S^{3} \simeq SU(2)$, the tangent space is parallelizable, and so, such metrics are in correspondence with metrics on $R^2$. Does such a characterization exist for $S^5$, or higher.

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Could you please clarify your second sentence? Note that $S^2=CP^1$. – Donu Arapura Sep 19 '11 at 16:32
Sorry, stupid mistake, fixed now. – Jean Delinez Sep 19 '11 at 16:56
Jean -- 1. one can't obtain the circle as the quotient of two $SU(n)$'s; 2. despite the fact that $S^3$ is parallelizable, there is just one invariant metric on it (up to multiplication by a positive real number); same for other spheres. – algori Sep 19 '11 at 17:10
@algori: $S^3$ is a Lie group, so it has as many left-invariant metrics there are inner products on the tangent space at the identity. Those inner products need not be invariant under the adjoint action, though. If they did you would get a bi-invariant metric on the group. – José Figueroa-O'Farrill Sep 19 '11 at 17:38
@Jean: why metrics on $R^2$? I would think "inner products on $R^3$" instead. – José Figueroa-O'Farrill Sep 19 '11 at 17:38
up vote 4 down vote accepted

Assume $n>2$. The $SU(n)$-invariant metrics on the sphere $S^{2n-1}$ are in bijection with $SU(n-1)$ invariant metrics on $T_x S^{2n-1}$ where $SU(n-1)$ is realized as the stabilizer of some $x\in S^{2n-1}$. This action is the sum of the defining $SU(n-1)$-module $V$ and the trivial real 1-dimensional module $T$. The space of invariant real valued bilinear forms on this is 2-dimensional and is spanned by the invariant Hermitian metric on the defining representation and some metric on the trivial representation.

Notice that if $(\cdot,\cdot)$ is invariant and $v\in V, t\in T$, then $(v,t)=0$ since there is an $A\in SU(n-1)$ such that $Av=-v$. This is what makes the case $n>2$ different from the case $n=2$.

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Yes, but is not the OP talking about $SU(n-1)$ invariant metrics? I agree that the question which has a simple answer is the one about $SU(n)$-invariant metrics, but I'm not sure that this is what was intended. – José Figueroa-O'Farrill Sep 19 '11 at 19:46
Yes, it was of course $SU(N)$ invariant metrics. This has been corrected now. – Jean Delinez Sep 19 '11 at 19:54
Am I correct in concluding that this implies that there is a unique bi-invariant metric up to scalar multiplication? – Jean Delinez Sep 19 '11 at 19:58
Jean -- when $n>2$, there are two linearly independent such metrics. One is the standard round metric; the other comes from the trivial representation of $SU(n-1)$; I don't know how to describe it geometrically. – algori Sep 19 '11 at 20:25

Classification of $SU(n+1)$-homogeneous metrics on $S^{2n+1}$:

First, note that for any $n\geq1$, the round metric on the sphere $S^{2n+1}$ can be scaled by $t^2$ in the direction of the Hopf fibers $S^1\to S^{2n+1}\to \mathbb C P^n$, giving rise to a one-parameter family $g_t$ of $SU(n+1)$-homogeneous metrics (so that $g_1$ is the original round metric).

Prop. Any $SU(n+1)$-homogeneous metric on $S^{2n+1}$ is isometric to some $g_t$, $t>0$.

In other words, they are always parameterized by (positive) real numbers. Alternatively, these $SU(n+1)$ homogeneous metrics on $S^{2n+1}$ are isometric to distance spheres in the complex projective space $\mathbb C P^{n+1}$.

Both of the above statements follow from a more general result, namely W. Ziller's classification of all homogeneous metrics on spheres, see [Homogeneous Einstein metrics on spheres and projective spaces. Math. Ann. 259 (1982), no. 3, 351–358].

A few more details:

An explicit formula for any $G$-homogeneous metric on the homogeneous space $G/H$, in terms of an $Ad(H)$-invariant decomposition $g= h\oplus p$ of the Lie algebra of $G$ is: $$\langle ,\rangle=h|_{p_0} + \sum_{i=1}^r\alpha_i B|_{p_i},$$ where $p= p_0\oplus\dots\oplus p_r$ is a decomposition so that the $H$ representations on $p_i$ are not equivalent, $H$ acts trivially on $p_0$ and irreducibly on $p_i$, $i=1,\dots r$; $h|_{p_0}$ is any inner product on $p_0$; $B$ is a bi-invariant metric on $G$ and $\alpha_i>0$ are real parameters. Any $G$-homogeneous metric is defined by choosing these parameters. In the case of $SU(n+1)$, this decomposition is $p=p_0\oplus p_1$ and $\dim p_0=1$, so (up to renormalization) there is only one parameter, $\alpha_1$ to be chosen (that I called $t$ above).

EDIT. Since I claim the metrics are parameterized by one positive real number (and another answer above claims there must be two real parameters), a clarification is in order here. The point is that, indeed, there are two parameters ($h|_{p_0}$ and $\alpha_1$ in the notation above), nevertheless it is always possible to divide the entire metric by the first one (the number that determines the metric $h$ on the 1-dim space $p_0$), which leaves us with just one parameter. Treating the family as having the $2$ parameters is slightly ambiguous because lots of metrics will be simple rescaling of other ones. In my description above, they are pairwise non-conformal.

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