Riemannian foliations on CROSSes

The problem of classifying Riemannian foliations on round spheres took a surprisingly long time to be fully solved, spanning several work of several people [Ran85], [GG88], [Wil01], [LW16]. In [LW16] the problem was finally solved, in fact for Riemannian foliations on topological spheres: Theorem 1.1 ([LW16]). Consider a k-dimensional Riemannian foliation F of a Riemannian manifold (M, g) homeomorphic to S n . Assuming 0 < k < n, one of the following holds:

(i) n = 2l + 1 for some l ∈ N >0 , k = 1 and the foliation is given by an isometric flow, up to changing the Riemannian metric. (ii) n = 4l + 3 for some l ∈ N >0 , k = 3 and the generic leaves are diffeomorphic to S 3 or RP 3 . (iii) n = 15, k = 7 and F is simple, given by the fibres of a Riemannian submersion (M, g) → (B, g B ) with (B, g B ) homeomorphic to S 8 and with fibres homeomorphic to S 7 . Furthermore all these cases can occur.

In this paper, we complete the classification of Riemannian foliations on manifolds homeomorphic to the remaining simply connected CROSSes, namely CP n , HP n and OP 2 : Theorem A. Consider a k-dimensional Riemannian foliation (M, F), 0 < k < dim M , of a Riemannian manifold (M, g) homeomorphic to a simply connected non-spherical CROSS. Then:

(1) M is homeomorphic to CP 2m+1 for some m ∈ N >0 and the foliation is given by the fibers of an

with its canonical metric, then the Riemannian submersion CP 2m+1 → B is congruent to the twistor bundle T : CP 2m+1 → HP m given by [x 0 : . . . :

In particular, no non-trivial Riemannian foliation can occur on manifolds homeomorphic to HP n or OP 2 .

Fix some Riemannian manifold (M, g) homeomorphic to CP n , and fix a homeomorphism f : M → CP n . The Hopf fibration H n : S 2n+1 → CP n pulls back to a “Hopf-like” principal S 1 -bundle Hn : f * (S 2n+1 ) → M :

) is homeomorphic to S 2n+1 via F , and we endow it with a metric which makes the S 1 -action isometric, and the map Hn a Riemannian submersion.

Assume there exists a Riemannian foliation (M, F) with k-dimensional fibres, with 0 < k < 2n. The idea is to pullback F to a foliation F := H-1 n (F) on f * (S 2n+1 ), and then use the classification given in [LW16].

Denote by L p the leaf of F through p ∈ M and by Lq the corresponding leaf in F through q ∈ H-1

By Theorem 1.1, dim( Lq ) = 3 or 7, meaning dim(L p ) = 2 or 6. We consider these cases separately, although with analogous methods.

From now on, we denote by L any leaf of F and let L = ( Hn ) -1 (L) ∈ F.

2.1. Case 1: F has 3-dimensional fibres. We know by [LW16] that, in this case, the generic leaf of F is diffeomorphic to S 3 or RP 3 . This means that any leaf L is covered by S 3 and in particular π 1 ( L) is a finite group. Furthermore, by [LW16] a 3-dimensional foliation can occur only on a sphere of dimension 4m + 3 for some m ∈ N >0 . Thus n = 2m + 1.

Note that the Hopf-like principal S 1 -bundle Hn : f * (S 4m+3 ) → M restricts to a principal S 1 -bundle S 1 → L → L. If L is diffeomorphic to S 3 , L is simply connected by the long exact sequence in homotopy of a fibration, and by the classification of compact and connected surfaces L is then diffeomorphic to S 2 (and thus obviously each fibre is closed and simply connected).

To get the same result when L is non-trivially covered by S 3 , we consider again the long exact sequence in homotopy of this fibration:

By the first part of this sequence, π 2 (L) ∼ = ker(ϕ) ̸ = 0 (since π 1 ( L) is finite): the only two-dimensional manifolds for which this is true are S 2 or RP 2 . Let us now try to exclude the RP 2 case. The fibration S 1 → L → L is orientable, so that we can consider its Gysin sequence:

or RP 2 , respectively. Then necessarily H 2 (L; Z) = Z and L is homeomorphic to S 2 and, in particular, simply connected. Since all leaves are simply connected, by Theorem 2.2 of [Esc82] the leaf space B := M/F is a Riemannian manifold and M → B, p → L p is a Riemannian submersion1 : F is simple.

Note that, given a Riemannian submersion π : N → M and a Riemannian foliation (M, F) whose leaves are given by fibres of some Riemannian submersion π ′ : M → B, then the leaves of the pullback foliation (N, π -1 (F)) are given by the fibres of π ′ • π, which is again a Riemannian submersion by composition. Therefore, the pullback of a simple Riemannian foliation is again simple, and

Thus in our case (f * (S 4m+3 ), F) is simple as well, given by the fibres of a Riemannian submersion f * (S 4m+3 ) → B. Since f * (S 4m+3 ) is a topological sphere, Theorem 5.1 of [Bro63] implies that L ≃ S 3 and (by the discussion in the introduction of [Bro63], or simply by considering the Gysin sequence of the fibration

The case of M isometric to CP 2m+1 with its canonical metric will be discussed later in Section 3. 2.2. Case 2: F has 7-dimensional fibres. We know that in this case the only possibility is 2n+1 = 15, so that n = 7. Proceeding like before, we get:

In this case, L is homeomorphic to S 7 and just like before S 1 → S 7 → L i

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