Riemannian Submersions: Understanding the Definitions and Well-Definedness

[Sajet]

Hi!

I have the following statements in a script on Riemannian submersions:

a) $T_{\bar p}\mathbb{CP}^n$ carries the structure of a complex vector space for any $\bar p \in \mathbb{CP}^n$.

b) We can associate $0 \neq v \in T_{\bar p}\mathbb{HP}^n$ with a four-dimensional subspace $v\mathbb H \subset T_{\bar p}\mathbb{HP}^n$.

($\pi$ is the submersion $\mathbb S^{2n+1} \rightarrow \mathbb{CP}^n$ or $\mathbb S^{4n+3} \rightarrow \mathbb{HP}^n$ respectively.)

Regarding a) it is then said: "Let $w \in T\mathbb{CP}^n, \lambda \in \mathbb C$. Let $\bar w$ be a horizontal lift of $w$. Define $\lambda w := \pi_*(\lambda \bar w)$. It is easily checked that this is well-defined."

I thought this was pretty clear. But then in b) they say:

"Let $w \in T\mathbb{HP}^n$, let $\bar w$ be a horizontal lift of w. Define $w\mathbb H := \pi_*(\bar w\mathbb H)$. It is also easily checked that this is well-defined.

Warning: For $\lambda \in \mathbb H$ we cannot set $w\lambda := \pi_*(\bar w\lambda)$ as this is not well-defined."

Now I don't see why exactly the last part is not well-defined. I thought the horizontal lift is unique, therefore $\bar w \lambda$ would be unique and $\pi_*(\bar w\lambda)$ as well.

Or maybe I just don't understand what well-defined means in either case, and why exactly this definition would be viable in a) but not in b).

I'd be very grateful if someone could help me understand this.

 

[quasar987]

The horizontal lift is unique only once a point in the fiber has been chosen.

That is, if p:M--N is a riemannian submersion and y is a point in N with v in TyN a tg vector at y, then to lift v, we need first to chose a point x in p-1(y), and then the horizontal lift of v to TxM is unique.

So they appear to be saying that for complex lambda, $\pi_*(\lambda\overline{w})$ is actually independant of the choice of x to lift too, but not in the case of quaternionic lambda.

But I don't even know what they mean by $\lambda \overline{w}$ in the complex case. Because the tg space of S^{2n+1}, being of odd dimension, cannot support a complex vector space structure...

 

[Sajet]

Wow, I completely ignored the fact that you first have to choose a point in the fiber in order to make the horizontal lift unique...

You're right that $T_p\mathbb S^{2n+1}$ does not carry a complex vector space structure, but the horizontal subspace $T_p^h\mathbb S^{2n+1} = (p\mathbb C)^\perp \cap \mathbb C^{n+1}$ does, and this is enough for this purpose.

Now I'm trying to see exactly why the multiplication is well-defined in a) and not in b).

Ok, I managed to construct a fairly complicated proof for a) but I hope there is an easier way:

Let $\bar w = \pi_{*q_1}(w_1) = \pi_{*q_2}(w_2), w_i \in T_{q_i}^h\mathbb S^{2n+1}$ horizontal lifts of $\bar w$. Proof that $\pi_{*q_1}(\lambda w_1) = \pi_{*q_2}(\lambda w_2)$

I know that $q_1, q_2$ are in the same fiber. Therefore $q_2 \in q_1S^1$.

I also know that $\pi_*(w_1) = \pi_*(w_2) \Rightarrow \|w_1\| = \|w_2\| \Rightarrow \|\lambda w_1\| = \|\lambda w_2\| =: r$

Now define $\tilde w_1 := \frac{\lambda w_1}{r}, \tilde w_2 := \frac{\lambda w_2}{r}$.

Now $\exp_{p}^{CP^n}(t\pi_{*q_1}(\lambda w_1)) = \exp(\pi_{*q_1}(t \cdot \tilde w_1 \cdot r)) = \pi(\exp_{q_1}(t\cdot \tilde w_1 \cdot r)) = ... = \cos(tr)q_1S^1+\sin(tr)\tilde w_1 S^1$

I can also show that $\tilde w_1 S^1 = \tilde w_2 S^1$. So the above equals

$\cos(tr)q_2 S^1+\sin(tr)\tilde w_2 S^1 = ... = \exp_p^{CP^n}(t\pi_{*q_2}(\lambda w_2))$

Because $\exp_p$ is injective in a neighborhood of 0 we get $\pi_{*q_1}(\lambda w_1) = \pi_{*q_2}(\lambda w_2)$

I want to give a presentation on this next week and I wouldn't want to make a fool of myself by proving something obvious in such a difficult way.

Do you happen to see if this can be proven much more quickly?

 

[quasar987]

Do you happen to see if this can be proven much more quickly?

If by "much more quickly" you mean without the need to compute, then I think I do! And as a bonus, we understand what fails in the quaternionic case.

First, a bit of notations. If a group G acts by riemannian isometries on (M,g), write $\pi:M\rightarrow M/G$ for the corresponding riemannian submersion, and write $\theta_g:M\rightarrow M$ for the map $p\mapsto g\cdot p$. Then $\pi=\pi\circ \theta_g$ for any g. Differentiating this relation gives $\pi_*=\pi_*\circ (\theta_g)_*$. Since $(\theta_g)_*$ is an isometric isomorphism, it preverses the horizontal and vertical subspaces. In particular, let p and q=gp be two points of M in the same G-orbit, let $w\in T_{[p]}(M/G)$, and let $\mathrm{Hor}_p(w)$, $\mathrm{Hor}_q(w)$ be the corresponding horizontal lifts of w above p and above q respectively. Then, $(\theta_g)_*(\mathrm{Hor}_p(w))=\mathrm{Hor}_q(w)$.

Now, in the case that interests us, M=S²ⁿ⁺¹, G=S¹, M/G=CPⁿ, and for a given p in S²ⁿ⁺¹, T_pS²ⁿ⁺¹ is naturally identified with $p^{\perp}\subset\mathbb{C}^{n+1}$. With this identification, $V_pS^{2n+1} = \mathrm{Ker}(\pi_*)=T_p(S^1\cdot p)$ is then naturally identified with $\mathbb{R}ip$, and so $H_pS^{2n+1}=(V_pS^{2n+1})^{\perp}$ is then naturally identified with $(ip)^{\perp}\cap p^{\perp}=\{p,ip\}^{\perp}$. As you noted, this is naturally a complex subspace of $\mathbb{C}^{n+1}$. Moreover, if q=up for some u in S¹ are two points in the same orbit, then I hold that $(\theta_u)_*:\{p,ip\}^{\perp}\rightarrow \{q,iq\}^{\perp}$ is just the map multiplication by u itself. To see this is the same trick that I explained to you in an earlier post: extend $\theta_u$ to a map Cⁿ⁺¹-->Cⁿ⁺¹. This is linear, so its derivative is just itself: multiplication by u. Now restrict back to the subspaces that interest you and you get that $(\theta_u)_*:\{p,ip\}^{\perp}\rightarrow \{q,iq\}^{\perp}$ is the map $\mathbf{v}\mapsto u\mathbf{v}$. In particular, it is a C-linear isomorphism: for any $\lambda\in\mathbb{C}$, $(\theta_u)_*(\lambda \mathbf{v})=u\lambda \mathbf{v}=\lambda u\mathbf{v}=\lambda (\theta_u)_*(\mathbf{v})$. This is why it makes sense to define a complex structure on CPⁿ by setting $\lambda w:=\pi_*(\lambda\mathrm{Hor}_p(w))$. Indeed, we have $\pi_*(\lambda\mathrm{Hor}_q(w))=\pi_*(\lambda (\theta_u)_*(\mathrm{Hor}_p(w)))=\pi_*( (\theta_u)_*(\lambda\mathrm{Hor}_p(w)))=\pi_*( \lambda \mathrm{Hor}_p(w))$.

Notice from the above computation that the C-linearity of $(\theta_g)_*$ is equivalent to the commutativity of the multiplication in C. So this is what fails in the quaternionic case! However, given any $u\in S^3\subset \mathbb{H}$ and any $\lambda \in \mathbb{H}$, there exists $\mu \in \mathbb{H}$ such that $(\theta_u)_*(\lambda \mathbf{v})=u\lambda \mathbf{v}=\mu u\mathbf{v}=\mu (\theta_u)_*(\mathbf{v})$. So while it does not make sense to define $\lambda w:=\pi_*(\lambda\mathrm{Hor}_p(w))$ in the quaternionic case, it does make sense to speak of $\mathbb{H}w$ as $\pi_*(\mathbb{H}\mathrm{Hor}_p(w))$ for any p.

 

[Sajet]

Wow, thank you, this helps me tremendously! This is clearly a much better and more insightful way of approaching this.