Curvature calculation of $S^{2n+1}=U(n+1)/U(n)$ as a homogeneous space

Question

I asked this question at StackExchange, but got no answer. So I am reposting it here.

I eventually want to check how the Hopf fibration $$ S^{2n+1}\to {\mathbb C}P^n $$ satisfies the Riemannian submersion formula, but I am frustrated that I could not even see how things work for $S^{2n+1}$ alone.

I can take either of the models $$ S^{2n+1} = U(n+1)/U(n) = SU(n+1)/SU(n+1), $$ but maybe the first is easier.

Then $G=U(n+1), H=U(n), {\mathfrak g}={\mathfrak u}(n+1), {\mathfrak h}={\mathfrak u}(n)$, diagonally embedded in ${\mathfrak u}(n+1)$ with the last entry 0. Let the invariant metric be $$ \langle A, B\rangle = -\frac{1}{2} \operatorname{Re Tr }AB = \frac{1}{2} \operatorname{Re Tr} AB^*.$$ Let $${\mathfrak m} = {\mathfrak h}^\perp, $$ whose dimension is $2n+1$. I am supposed to check that for any orthonormal $X, Y\in {\mathfrak m}$, the sectional curvature $$ K(X, Y) = 1. $$

Note that for the ${\mathbb C}P^n=SU(n+1)/U(n)$ case, the $K$ ranges from 1 to 4, and I can see that in this calculation, depending on if $X$ and $Y$ are in a complex subspace.

Then I don't see how it works out in the $S^{2n+1}$ case, as we should always have $K$ as a constant 1. I undestand that in this case, $[{\mathfrak m}, {\mathfrak m}]\not\subset {\mathfrak h}$, so I need to use the more general formula (Cheeger & Ebin, Cor. 3.33) $$ K(X, Y) = \frac{1}{4} \|[X, Y]_{\mathfrak m}\|^2 + \|[X, Y]_{\mathfrak h}\|^2, $$ but it still would not work for me.

If $X, Y$ are in a complex subspace, say \begin{align} X&=E_{1, n+1}-E_{n+1, 1}, \\ Y&=i(E_{1, n+1}+E_{n+1, 1}), \text{ then}\\ [X, Y] &= 2iE_{1, 1} - 2iE_{n+1,n+1}. \end{align} I would get $2\frac{1}{2}$ as my answer.

I know I may be too obsessed, and this could be a normalization issue, but your help would be appreciated.

Answer 1

As pointed out by @emiliocba, the metric I am considering does not have constant sectional curvature, so there was nothing wrong with my calculation, and it actually perfectly matches with what should happen.

I am led to read a bit more about Berger spheres and works of Ziller and his group. Here is what I found.

Consider the Hopf fibration $S^{2n+1}\to {\mathbb C}P^n$ from the round sphere with sectional curvature 1 to the complex projective space with the Fubini-Study metric. Then we have a decomposition of the metric $$ g = g_{\mathcal V} \oplus g_{\mathcal H}, $$ along the fiber $S^1$ and its orthogonal complement. A Berger sphere has a deformed metric $$ g_t = t g_{\mathcal V} \oplus g_{\mathcal H}, \quad t>0, $$ with the fiber circles scaled. The canonical round sphere corresponds to $t=1$.

When I consider the homogeneous space $$ S^{2n+1} = U(n+1)/U(n), $$ induced from the biinvariant metric on $U(n+1)$, I think the metric turns out to be $$ g_{\frac{1}{2}}. $$ This can be seen by considering $$ Z= iE_{n+1,n+1}, $$ which has $\|Z\|^2 = \frac{1}{2} \operatorname{Re Tr} Z^* Z = \frac{1}{2}$, while its image on the round sphere should have norm squared 1. For this Berger sphere $g_{\frac{1}{2}}$, the maximal sectional curvature is $\frac{5}{2}$ as I computed as $K(X, Y)$ in the original post. The minimal sectional curvature is $\frac{1}{2}$, and can be realized as $K(X, \sqrt{2} Z)$, since \begin{align} [X, \sqrt{2}Z] &= \sqrt{2}Y\in {\mathfrak m},\\ K(X, \sqrt{2}Z) &= \frac{1}{4}\|[X, \sqrt{2}Z]\|^2 = \frac{1}{2}. \end{align}

Such max and min sectional curvatures match formulas proved in general in Verdiani, Ziller, Math Z (2009), p. 477.

If we consider $$ S^{2n+1} = SU(n+1)/SU(n), $$ then the induced metric is stated in Grove and Ziller, Invent. Math. (2002) Table 2.4. as $$ g_{\frac{n+1}{2n}}, $$ as can be seen by analyzing the element $W=\operatorname{diag}(-i, \cdots, -i, ni)$. One can also analyze the possible max and min sectional curvatures there.

As to my original goal of understanding the submersion formula for the Hopf fibration, it almost becomes tautological now. In general, it is well studied as an example of \begin{align} H\subset K\subset G;&\quad K/H\to G/H\to G/K,\\ SU(n)\subset U(n)\subset SU(n+1);&\quad S^1\to S^{2n+1}\to {\mathbb C}P^n, \end{align} and all curvatures are computed from that of $G$ using the submersion formula. Hence they are compatible.