Cayley fibrations in the Bryant–Salamon Spin(7) manifold

On each complete asymptotically conical Spin(7)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\mathrm{Spin}\,}}(7)$$\end{document} manifold constructed by Bryant and Salamon, including the asymptotic cone, we consider a natural family of SU(2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\mathrm{SU}\,}}(2)$$\end{document} actions preserving the Cayley form. For each element of this family, we study the (possibly singular) invariant Cayley fibration, which we describe explicitly, if possible. These can be reckoned as generalizations of the trivial flat fibration of R8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^8$$\end{document} and the product of a line with the Harvey–Lawson coassociative fibration of R7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^7$$\end{document}. The fibres will provide new examples of asymptotically conical Cayley submanifolds in the Bryant–Salamon manifolds of topology R4,R×S3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^4, {\mathbb {R}}\times S^3$$\end{document} and OCP1(-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}}_{{\mathbb {C}}{\mathbb {P}}^1} (-1)$$\end{document}.


Introduction
In 1926, Cartan showed how to associate a group to any Riemannian manifold through parallel transport [Car26].He called such a group the holonomy group of the Riemannian manifold, and he used it to classify symmetric spaces.Almost 30 years later, Berger found all the groups that could appear as the holonomy of a simply-connected, nonsymmetric, and irreducible Riemannian manifold [Ber55].The exceptional holonomy groups G 2 and Spin(7) belonged to this list.The existence of Riemannian manifolds with such holonomy was unknown until Bryant [Bry87] provided incomplete examples and Bryant-Salamon [BS89] provided complete ones.In particular, Bryant and Salamon constructed a 1-parameter family of torsion-free G 2 -structures on Λ 2 − (T * S 4 ), Λ 2 − (T * CP 2 ), S /(S 3 ), and a 1-parameter family of torsion-free Spin(7)-structures on S / − (S 4 ).The holonomy principle implies that the holonomy group of these manifolds is contained in G 2 and Spin(7), respectively.As Bryant and Salamon proved that their examples have full holonomy, the problem of the classification of Riemannian holonomy groups is settled.
Manifolds with exceptional holonomy are Ricci-flat and admit natural calibrated submanifolds.These are the associative 3-folds and the coassociative 4-folds in the G 2 case, while they are the Cayley 4-folds in the Spin(7) one.A crucial aspect of the study of manifolds with exceptional holonomy regards fibrations through these natural submanifolds.One of the main reasons for the interest in calibrated fibrations comes from mathematical physics.Indeed, analogously to the SYZ conjecture [SYZ96], that relates special Lagrangian fibrations in mirror Calabi-Yau manifolds, one would expect similar dualities for coassociative fibrations in the G 2 case and Cayley fibrations in the Spin(7) one.We refer the reader to [GYZ03,Ach98] for further details.Another reason lies in the attempt to understand and construct new compact manifolds with exceptional holonomy through these fibrations [Don17].
Some work has been carried out in the G 2 case (see f.i.[ABS20,Bar10,Don17,KLo21,Li19]), while little is known in the Spin(7) setting.In particular, Karigiannis and Lotay [KLo21] constructed an explicit coassociative fibration on each G 2 Bryant-Salamon manifold and the relative asymptotic cone.To do so, they chose a 3-dimensional Lie group acting through isometries preserving the G 2 -structure, and they imposed the fibres to be invariant under this group action.In this way, the coassociative condition is reduced to a system of tractable ODEs defining the fibration.Previously, this idea was used to study cohomogeneity one calibrated submanifolds related to exceptional holonomy in the flat case by Lotay [Lot07] and in Λ 2 − (T * S 4 ) by Kawai [Kaw18].Analogously, we consider Cayley fibrations on each Spin(7) Bryant-Salamon manifold and the relative asymptotic cone, which are invariant under a natural family of structure-preserving SU(2) actions.The first key observation, due to Bryant and Salamon [BS89], is that Sp(2) × Sp(1) is contained in the subgroup of the isometry group that preserves the Spin(7)-structure.Indeed, one can lift an action of SO(5) on S 4 to an action of Spin(5) ∼ = Sp(2) on the spinor bundle of S 4 .The Sp(1) factor of Sp(2) × Sp(1) comes from a twisting of the fibre.Clearly, this group admits plenty of 3-dimensional subgroups.The family we consider consists of the subgroups that respect the direct product, i.e. that do not sit diagonally in Sp(2) × Sp(1).Through Lie group theory, it is easy to find these subgroups.Indeed, they either are the whole Sp(1), appearing in the second factor or the lift of one of the following subgroups of SO(5), which are going to be contained in the first factor: where Sp(1) × Id 1 denotes both the subgroup acting on H × R by left multiplication and by right multiplication of the quaternionic conjugate.Observe that their lifts to Sp(2) are all diffeomorphic to SU(2) ∼ = Sp(1).Moreover, the Sp(1) contained in the second factor will only act on the fibres of S / − (S 4 ), leaving the base fixed.
Summary of results and organization of the paper.In section 2, we briefly review some basic results on Spin(7) and Riemannian geometry.In particular, once fixed the convention for the Spin(7)-structure, we recall the definition of Cayley submanifolds, together with Karigiannis-Min-Oo's characterization [KM05, Proposition 2.5], and Cayley fibrations.Similarly to [KLo21, Definition 1.2], our notion of Cayley fibrations allows the fibres to be singular and to self-intersect.Finally, we provide the definitions of asymptotically conical and conically singular manifolds.Section 3 contains a detailed description of the 1-parameter family of Spin(7) manifolds constructed by Bryant-Salamon.Here, we also discuss the automorphism group.In particular, we briefly explain why the system of ODEs characterizing the fibration induced by the irreducible action of SO(3) on S 4 is going to be too complicated to be solved.
Starting from section 4, we deal with Cayley fibrations.Here, we study the fibration invariant under the SU(2) acting only on the fibres of S / − (S 4 ).In this case, the fibration is trivial, i.e. coincide with the usual projection map from S / − (S 4 ) to S 4 .We compute the multi-moment map in the sense of [MS12,MS13], which is a polynomial depending on the square of the distance function.Blowing-up at any point of the zero section, the fibration becomes the trivial flat fibration of R 8 .
In section 5, we consider the action on S / − (S 4 ) induced by SO(3) × Id 2 ⊂ SO(5) acting on S 4 .Under a suitable choice of metric-diagonalizing coframe on an open, dense set U, the system of ODEs characterizing the Cayley condition is completely integrable, and hence we obtain a locally trivial fibration on U whose fibres are Cayley submanifolds.Extending by continuity the fibration to the whole S / − (S 4 ), we prove that the parameter is S 4 and the fibres are topological R 4 s, O CP 1 (−1)s or R × S 3 s.Through a asymptotic analysis, it is easy to see that the R 4 s separating the Cayleys of different topology are the only singular ones.The singularity is asymptotic to the Lawson-Osserman cone [LO77].Each Cayley intersects at least another one in the zero section of S / − (S 4 ), and, at infinity, they are asymptotic to a non-flat cone with link S 3 endowed with either the round metric or a squashed metric.While in the G 2 case [KLo21, Subsection 5.7, Subsection 6.7], the multi-moment map they explicitly compute has a clear geometrical interpretation, it does not in our case.Finally, keeping track of the Cayley fibration, blowing-up at the north pole, we obtain the fibration on R 8 , which is given by the product of the SU(2)-invariant coassociative fibration constructed by Harvey and Lawson [HL82, Section IV.3] with a line.
We deal with the Cayley fibration invariant under the SU(2) action induced from Sp(1) × Id 1 in Section 6.The left quaternionic multiplication gives the same fibration as the conjugate right quaternionic multiplication up to orientation.Contrary to the previous case, we can not completely integrate the system of ODEs we obtain on an open, dense set U. However, we deduce all the information we are interested in via a dynamical system argument.In particular, we show that the fibres are parametrized by a 4-dimensional sphere and that they are smooth submanifolds of topology S 3 × R, R 4 or S 4 .The unique point of intersection is the south (north) pole of the zero section, where all fibres of topology R 4 and the sole Cayley of topology S 4 (i.e. the zero section) intersect.It is easy to show that all Cayleys are asymptotic to a non-flat cone with round link S 3 .We also compute the multi-moment map, and show that the fibration converges to the trivial flat fibration of R 8 when we blow-up at the north pole.
The last group action that would be natural to study is the lift of SO(3) acting irreducibily on R 5 .However, in this case the ODEs become extremely complicated and can not be solved explicitely.Moreover, the analogous action on the flat Spin(7) space and on the Bryant-Salamon G 2 manifold Λ 2 − (T * S 4 ) was studied by Lotay [Lot05, Subsection 5.3.3] and Kawai [Kaw18], respectively.In both cases, the defining ODEs for Cayley submanifolds and coassociative submanifolds were too complicated.
Acknowledgements.The author wishes to thank his supervisor Jason D. Lotay for suggesting this project and for his enormous help and guidance.He also wishes to thank the referee for carefully reading an earlier version of this paper and for greatly improving its exposition.This work was supported by the Oxford-Thatcher Graduate Scholarship.

Preliminaries
In this section, we recall some basic results concerning Spin(7) manifolds, Cayley submanifolds and Riemannian conifolds.
Remark 2.1.This choice of convention for Φ R 8 is compatible with the fact that we will be working on S / − (S 4 ).Indeed, we can identify our local model with S / − (R 4 ).Further details regarding the sign conventions and orientations for Spin(7)-structures can be found in [Kar10].
Definition 2.2.Let M be a manifold and let Φ be a 4-form on M .We say that Φ is admissible if, for every x ∈ M , there exists an oriented isomorphism i x : R 8 → T x M such that i * x Φ = Φ R 8 .We also refer to Φ as a Spin(7)-structure on M .
The Spin(7)-structure on M also induces a Riemannian metric, g Φ , and an orientation, vol Φ , on M .With respect to these structures Φ is self-dual.We refer the reader to [SW17] for further details.

2.2.
Cayley submanifolds and Cayley fibrations.Given (M, Φ), Spin(7) manifold, it is clear that Φ has comass one, and hence, it is a calibration.Definition 2.4.We say that a 4-dimensional oriented submanifold is Cayley if it is calibrated by Φ, i.e., if Φ N 4 = vol N 4 .Fixed a point p ∈ M , a 4-dimensional oriented vector subspace H of T p M is said to be a Cayley 4-plane if Φ p calibrates H.
Remark 2.5.Observe that N is a Cayley submanifold if and only if T p N is a Cayley 4-plane for all p ∈ N .
We now give Karigiannis and Min-Oo characterization of the Cayley condition.
Proposition 2.6 (Karigiannis-Min-Oo [KM05, Proposition 2.5]).The subspace spanned by tangent vectors u, v, w, y is a Cayley 4-plane, up to orientation, if and only if the following form vanishes: where B(u, v, w) := w v u Φ and Remark 2.7.The reduction of the structure group of M to Spin(7) induces an orthogonal decomposition of the space of differential k-forms for every k, which corresponds to an irreducible representation of Spin(7).In particular, if k = 2, the irreducible representations of Spin(7) are of dimension 7 and 21.At each point x ∈ M , these representations induce the decomposition of Λ 2 (T * x M ) into two subspaces, which we denote by Λ 2 7 and Λ 2 21 , respectively.The map π 7 defined in Proposition 2.6 is precisely the projection map from the space of two-forms to Λ 2 7 .Further details can be found in [SW17].
Following [KLo21], we extend the definition of Cayley fibration so that it may admit intersecting fibres and singular fibres.Remark 2.9.The last point allows the Cayley submanifolds in the family B to intersect.Indeed, this may happen in M \ M .Moreover, we may lose information (e.g.completeness and topology) when we restrict the Cayley fibres to M .
We conclude this subsection explaining how we determine the topology of R 2 bundles over S 2 ∼ = CP 1 arising as the smooth fibres of a Cayley fibration.This is the same discussion used in [KLo21].Let N be the total space of an R 2 -bundle over CP 1 which is also a Cayley submanifold of a Spin(7) manifold (M, Φ).Since N is orientable and it is the total space of a bundle over an oriented base, it is an orientable bundle.We deduce that N is homeomorphic to a holomorphic line bundle over CP 1 .These objects are classified by an integer k ∈ Z and are denoted by O CP 1 (k).Moreover, for k > 0 we have the following topological characterization of O CP 1 (−k): In the situation we will consider, the submanifolds we construct have the form N \ S 2 = R + × S 3 .Hence, the only possibility is to obtain topological O CP 1 (−1)s.

Riemannian conifolds.
We now recall the definitions of asymptotically conical and conically singular Riemannian manifolds.
Definition 2.10.A Riemannian cone is a Riemannian manifold (M 0 , g 0 ) with M 0 = R + × Σ and g 0 = dr 2 + r 2 g Σ , where r is the coordinate on R + and g Σ is a Riemannian metric on the link of the cone, Σ.
Definition 2.11.We say that a Riemannian manifold (M, g) is asymptotically conical (AC) with rate λ < 0 if there exists a Riemannian cone (M 0 , g 0 ) and a diffeomorphism Ψ : (R, ∞)×Σ → M \K satisfying: where K is a compact set of M and R > 0. (M 0 , g 0 ) is the asymptotic cone of (M, g) at infinity.Definition 2.12.We say that a Riemannian manifold (M, g) is conically singular with rate µ > 0 if there exists a Riemannian cone (M 0 , g 0 ) and a diffeomorphism Ψ : (0, ) × Σ → M \ K satisfying: where K is a closed subset of M and > 0. (M 0 , g 0 ) is the asymptotic cone of (M, g) at the singularities.
Remark 2.13.As Σ does not need to be connected, AC manifolds may admit more than one end and asymptotically singular manifolds may admit more than one singular point.

Bryant-Salamon Spin(7) manifolds
In this section we will describe the central objects of this work, i.e., the Spin(7) manifolds constructed by Bryant and Salamon in [BS89].There, they provided a 1-parameter family of torsion-free Spin(7)-structures on M := S / − (S 4 ), the negative spinor bundle on S 4 .The 4-dimensional sphere is endowed with the metric of constant sectional curvature k, which is the unique spin self-dual Einstein 4-manifold with positive scalar curvature [Hit81].Without loss of generality, we rescale the sphere so that k = 1.
Remark 3.1.The Bryant-Salamon construction on S 4 also works on spin 4-manifolds with selfdual Einstein metric, but negative scalar curvature, and on spin orbifolds with self-dual Einstein metric.However, in these cases, the metric is not complete or smooth.
3.1.The negative spinor bundle of S 4 .Let S 4 be the 4-sphere endowed with the Riemannian metric of constant sectional curvature 1.As S 4 is clearly spin, given P SO(4) frame bundle of S 4 we can find the spin structure P Spin(4) together with the spin representation: where µ ± (p ± )(v) := vp ± .Let π : P Spin(4) → P SO(4) be the double cover in the definition of spin structure, and let πn 0 : Spin(n) → SO(n) be the double (universal) covering map for all n ≥ 3. The negative spinor bundle over S 4 is defined as the associated bundle: The positive spinor bundle is defined analogously, taking µ + instead.
Given an oriented local orthonormal frame for S 4 , {e 0 , e 1 , e 2 , e 3 }, the real volume element e 0 • e 1 • e 2 • e 3 acts as the identity on the negative spinors and as minus the identity on the positive ones.Now, let {b 0 , b 1 , b 2 , b 3 } be the dual coframe of {e 0 , e 1 , e 2 , e 3 }, let ω the connection 1-forms relative to the Levi-Civita connection of S 4 with respect to the frame {e 0 , e 1 , e 2 , e 3 } and let {σ 1 , σ i , σ j , σ k } a local orthonormal frame for the negative spinor bundle corresponding to the standard basis of {1, i, j, k} in this trivialization.Hence, we can define the linear coordinates (a 0 , a 1 , a 2 , a 3 ) which parametrize a point in the fibre as a 0 σ 1 + a 1 σ i + a 2 σ j + a 3 σ k .
By the properties of the spin connection and the fact we are working on the negative spinor bundle, we can write: It is well-known that these are the connection forms on the bundle of anti-self-dual 2-forms, with respect to the connection induced by the Levi-Civita connection on S 4 and the frame given by is a cyclic permutation of (1, 2, 3).The ρ i s are characterized by: and the vertical one forms are: Remark 3.2.If we denote by π S 4 the vector bundle projection map from S / − (S 4 ) to S 4 , we can obtain horizontal forms on S / − (S 4 ) via pull-back.For example, {π * S 4 (b i )} 4 i=1 and the linear combinations of their wedge product are horizontal forms on S / − (S 4 ).In order to keep our notation light, we will omit the pullback from now on.
As S 4 is self-dual and with scalar curvature equal to 12k, we have: , which is equivalent to [BS89, p. 842] and [KLo21,3.24].We can use it to compute: (3.3) that is going to be useful below.
Remark 3.3.A detailed account of spin geometry can be found in [LM89].Observe that, there, the definition of positive and negative spinors is interchanged.We opted to stay consistent with [BS89].Indeed, the vertical 1-forms we obtain coincide with the ones obtained by Bryant and Salamon, up to renaming the ρ i s.The same holds for the relative exterior derivatives.
3.2.The Spin(7)-structures.If r 2 := a 2 0 + a 2 1 + a 2 2 + a 2 3 is the square of the distance function from the zero section and c is a positive constant, then, the Spin(7)-structures defined by Bryant and Salamon are: where The metric induced by Φ c is while the induced volume element is , we obtain a Spin( 7) cone (M 0 , Φ 0 ), i.e.M 0 with the metric induced by the Spin(7)-structure Φ 0 is a Riemannian cone.
It is well-known that the Bryant-Salamon Spin(7) manifolds we have just described are asymptotically conical (see for instance [Sal89, p.184]), hence, we state here the main results concerning their asymptotic geometry.

Automorphism Group.
A natural subset of the diffeomorphism group of a Spin(7)-manifold is the automorphism group, i.e. the subgroup that preserves the Spin(7)-structure.Clearly, the automorphism group is contained in the isometry group with respect to the induced metric.
In the setting we are considering, Bryant and Salamon noticed that the diffeomorphisms given by the Sp(2)×Sp(1)-action described as follows are actually in the automorphism group [BS89, Theorem 2].Consider SO(5) acting on S 4 in the standard way.This induces an action on the frame bundle of S 4 via the differential, which easily lifts to a Spin(5) ∼ = Sp(2) action on P Spin(4) .If we combine it with the standard quaternionic left-multiplication by unit vectors on H, we have defined an Sp(2) × Sp(1) action on P Spin(4) × H.As it commutes with µ − , it passes to the quotient S / − (S 4 ).
By Lie group theory [Kaw18, Appendix B], we know that the 3-dimensional connected closed subgroups of Sp(2) are the lift of one of the following subgroups of SO(5): where Sp(1) × Id 1 denotes both the subgroup acting on H × R by left multiplication and by right multiplication of the quaternionic conjugate.Observe that they are all diffeomorphic to SU(2).In particular, the family of 3-dimensional subgroups that do not sit diagonally in where G is one of the lifts above.These are going to be the subgroups of the automorphism group that we will take into consideration.

The Cayley fibration invariant under the Sp(1) action on the fibre
Let M := S / − (S 4 ) and M 0 := R + × S 7 endowed with the torsion-free Spin(7)-structures Φ c constructed by Bryant and Salamon and described in Section 3.
Observe that (M, Φ c ) and (M 0 , Φ 0 ) admit a trivial Cayley Fibration.Indeed, it is straightforward to see that the natural projection to S 4 realizes both spaces as honest Cayley fibrations with smooth fibres diffeomorphic to R 4 and R 4 \ {0}, respectively.In both cases, the parametrizing family is clearly S 4 The fibres are asymptotically conical to the cone of link S 3 and metric: where s = r 3/5 10/3 and g S 3 is the standard unit round metric.Since Id Sp(2) × Sp(1) acts trivially on the basis, and as Sp(1) on the fibres of S / − (S 4 ) identified with H, it is clear that the trivial fibration is invariant under Id Sp(2) × Sp(1).
Remark 4.1.We compute the associated multi-moment map, ν c , in the sense of Madsen and Swann [MS12,MS13].This is: where we subtracted c 6/5 100/3 so that the range of the multi-moment map is [0, ∞).Observe that the level sets of ν c coincide with the level sets of the distance function from the zero section.Let M := S / − (S 4 ) and M 0 := R + × S 7 be endowed with the torsion-free Spin(7)-structures Φ c constructed by Bryant and Salamon that we described in Section 3. On each Spin(7) manifold, we construct the Cayley Fibration which is invariant under the lift to M (or M 0 ) of the standard 5.1.The choice of coframe on S 4 .As in [KLo21], we choose an adapted orthonormal coframe on S 4 which is compatible with the symmetries we will impose.Since the action coincides, when restricted to S 4 , with the one used by Karigiannis and Lotay on Λ 2 − (T * S 4 ) [KLo21, Section 5], it is natural to employ the same coframe, which we now recall.
We split R 5 into the direct sum of a 3-dimensional vector subspace P ∼ = R 3 and its orthogonal complement P ⊥ ∼ = R 2 .As S 4 is the unit sphere in R 5 , we can write, with respect to this splitting: Now, for all (x, y) ∈ S 4 there exists a unique α ∈ [0, π/2], some u ∈ S 2 ⊂ P and some v ∈ S 1 ⊂ P ⊥ such that: Observe that u and v are uniquely determined when α ∈ (0, π/2), while, when α = 0, π/2, v can be any unit vector in P ⊥ (y = 0) and u can be any unit vector in P (x = 0), respectively.Hence, we are writing S 4 as the disjoint union of an S 2 , corresponding to α = 0, of an S 1 , corresponding to α = π/2, and of S 2 × S 1 × (0, π/2).

The horizontal and the vertical space.
As in [KLo21, Subsection 5.2], we use (3.1) to compute the ρ i 's in the coordinate frame we have just defined.Indeed, (5.1) implies that: (5.2) hence, we deduce that: We conclude that in these coordinates we have: Now that we have computed the connection forms, we immediately see from (3.2) that the vertical one forms are: (5.3) 5.3.The SU(2) action.Given the splitting of subsection 5.1, R 5 = P ⊕ P ⊥ , since P ∼ = R 3 and P ⊥ ∼ = R 2 , we can consider SO(3) acting in the usual way on P and trivially on P ⊥ .In other words, we see SO(3) ∼ = SO(P ) × Id P ⊥ ⊂ SO(P ⊕ P ⊥ ) ∼ = SO(5).Obviously, this is also an action on S 4 .By taking the differential, SO(3) acts on the frame bundle P SO(4) of S 4 .The theory of covering spaces implies that this action lifts to a Spin(3) ∼ = SU(2) action on the spin structure P Spin(4) of S 4 .In particular, the following diagram is commutative: (5.4) .
Finally, if Spin(3) acts trivially on H, we can combine the two Spin(3) actions to obtain one on P Spin(4) × H, which descends to the quotient P Spin(4) × µ − H = S / − (S 4 ).
Remark 5.1.Recall that T S 4 = P SO(4) × • R 4 , where • is the standard representation of SO(4) on R 4 .Let G be a subgroup of SO(5) which acts on P SO(4) × • R 4 via the differential on the first term and trivially on the second.It is straightforward to verify that this action passes to the quotient and that it coincides with the differential on T S 4 .Now, we describe the geometry of this Spin(3) action on S / − (S 4 ).Since π is fibre-preserving and (5.4) represents a commutative diagram, we observe that, fixed a point p = (x, y) ∈ S 4 ⊂ P ⊕ P ⊥ , the subgroup of Spin(3) that preserves the fibre of P Spin(4) over p is the lift of the subgroup of SO(3) that fixes the fibre of P SO(4) over p.
Using once again the commutativity of (5.4) and Claim 1, we deduce that the action in the trivialization of S / − (S 4 ) induced by {e i } 3 i=0 is as follows: , where ĥγ := cos(γ/2) − i sin(γ/2) and where a ∈ H.
Claim 2. Spin(3) acts on the fibre of S / − (S 4 ) as Sp(1) acts on H via right multiplication of the quaternionic conjugate.
We put all these observations in a lemma.
Table 1.Spin(3) Orbits 5.4.Spin(3) adapted coordinates.The description of the SU(2) action that we carried out in Subsection 5.3 suggests the following reparametrization of the linear coordinates (a 0 , a 1 , a 2 , a 3 ) on the fibres of S / − (S 4 ): where s, t ∈ [0, ∞), γ ∈ [0, 4π) and δ ∈ [0, 2π).This is a well-defined coordinate system when s and t are strictly positive; we will assume this from now on.Geometrically, γ represents the SU(2) action, while δ can be either seen as the phase in the orbit of the action when (a 0 , a 1 ) = (s, 0) or as twice the common angle in [0, π) that the suitable point in the orbit makes with (s, 0) and (t, 0).These interpretations can be recovered by putting γ = δ and γ = 0, respectively.
Lemma 5.4.The vertical 2-forms A 1 , A 2 , A 3 , in the adapted frame defined in Subsection 5.4, have the form: (5.9) Proof.Computing the exterior derivatives of the a i 's in the coordinates (5.6), we can reduce our statement to a long computation based on (5.3).
Remark 5.6.Using the identities: (5.13) (5.14) one could easily find Φ c in the adapted frame of Subsection 5.4.It is clear from Corollary 5.5 that it is not going to be in a nice form.
Lemma 5.7.Given c ≥ 0, the Riemannian metric g c , in the adapted frame of Subsection 5.4, takes the form: where Proof.Combining (3.5), (5.1), (5.3) and (5.7), it is easy to obtain the Riemannian metric in the claimed form.
Proof.The first addendum remains invariant from Lemma 5.7, while (5.15) implies that the remaining part is equal to the second addendum in Lemma 5.7.
In particular, using this coframe, we sacrifice compatibility with the group action to obtain a simpler form for Φ c and a diagonal metric.
Proof.It is straightforward to verify these identities from (5.15) and the definition of dual frame.5.7.The Cayley condition.As the generic orbit of the SU(2) action we are considering is 3dimensional (see Lemma 5.2), it is sensible to look for SU(2)-invariant Cayley submanifolds.Indeed, Harvey and Lawson theorem [HL82, Theorem IV .4.3] guarantees the local existence and uniqueness of a Cayley passing through any given generic orbit.To construct such a submanifold N , we consider a 1-parameter family of 3-dimensional SU(2)-orbits in M .Hence, the coordinates that do not describe the orbits, i.e. α, β, s, t and δ, need to be functions of a parameter τ .Explicitly, we have: (5.22) and its tangent space is spanned by: {∂ 1 , ∂ 2 , ∂ 3 , ṡ∂ s + ṫ∂ t + α∂ α + β∂ β + δ∂ δ }, where the dots denotes the derivative with respect to τ .The Cayley condition imposed on this tangent space (see Proposition 2.6) generates a system of ODEs on α, β, s, t, δ.
As it mainly consists of computations, we leave the proof of Theorem 5.12 to Appendix A.
Corollary 5.13.Let N be an SU(2)-invariant submanifold as described at the beginning of this subsection.Then, N is Cayley in the chart U if and only if the following system of ODEs is satisfied: , where r 2 = s 2 + t 2 as usual.
Proof.As α ∈ (0, π/2) and s, t > 0, we get immediately the first three equations from the first three equations of (5.23).The last two equations of (5.23) are superfluous as β = 0 and δ = 0.The same holds for t times the fourth equation plus s times the fifth equation of (5.23), where we use t ṡ − s ṫ = 0 this time.We conclude by considering s times the fifth equation minus t times the fourth equation of (5.23).
5.8.The Cayley fibration.In the previous section we found the condition that makes N , SU(2)invariant submanifold, a Cayley submanifold.Explicitly, it consists of a system of ODEs that is completely integrable; these solutions will give us the desired fibration.
Theorem 5.14.Let N be an SU(2)-invariant submanifold as described at the beginning of Subsection 5.7.Then, N is Cayley in U if and only if the following quantities are constant: where H(α) is the primitive function of h(α) := (cos α sin α) 3/2 .
Proof.The condition on β and δ follows immediately from Corollary 5.13.Taking the derivative in τ of s/t, we see that , which is equivalent to the second equation in Corollary 5.13, as t > 0. Analogously, one can see that the derivative with respect to τ of F is equivalent to the last equation of Corollary 5.13 if we assume that s/t is constant.

Setting
the preserved quantities transform to: where we multiplied F by the constant (v 2 + 1).Observe that this is an admissible transformation from s, t ∈ (0, ∞) to u, v ∈ (0, ∞).Moreover, fixed β, δ, v, we can represent the SU(2)-invariant Cayley submanifolds as the level sets of F reckoned as a R-valued function of α and u.An easy analysis of F shows that these level sets can be represented as in Figure 1.The dashed lines in the two graphs correspond to the curves formed by the u-minimums of each level set and to the two vertical lines: α = arccos(1/ √ 6).For c = 0, these coincide, while in the generic case the locus of the u-minimum is: The conical version.We first consider the easier case, i.e. when c = 0.It is clear from the graph that the SU(2)-invariant Cayleys passing through U are contained in U, have topology S 3 × R and are smooth.Moreover, we can construct a Cayley fibration on the chart U with base an open subset of R 4 .To do so, we associate to each point of U the value of β, δ, s/t and F of the Cayley passing through that point.This SU(2)-invariant fibration naturally extends to the whole M 0 via continuity.Using Table 1 and Harvey and Lawson uniqueness theorem [HL82, Theorem IV.4.3], we can describe the extension precisely.Indeed, when α = π/2, the fibres of π S 4 are SU(2)-invariant Cayley submanifolds; when α = π/2 and s = 0 or t = 0, the suitable Cayley submanifolds constructed by Karigiannis and Min-Oo [KM05] are SU(2)-invariant; finally, when The smooth version.Now, we consider the generic case, i.e. when c > 0. Differently from the cone, the graph of the level sets of F shows that the SU(2)-invariant Cayley submanifolds passing through U do not remain contained in it, and they admit three different topologies in the extension.The red, black and blue lines correspond to submanifolds with topology R × S 3 , R 4 and O CP 1 (−1), respectively.Indeed, the first two cases are obvious, while, if we assume smoothness, we can deduce the third one through the argument of Subsection 2.2.We define an SU(2)-invariant Cayley fibration on U that extends to the whole M exactly as above.If we fix a value of F corresponding to a Cayley of topology O CP 1 (−1), then, for every δ, v, all the different Cayleys will intersect in a CP 1 ⊂ S 4 , where S 4 is the zero section of S / − (S 4 ).In particular, the M of Definition 2.8 is equal to M 0 in this context, i.e., we can assume u > 0.
The parametrizing space.Using Figure 1, we can study the parametrizing space B of the Cayley fibrations we have just described.We will only deal with the smooth version, as the conical case is going to be completely analogous.
Ignoring β for a moment, it is immediate to see that, if we restrict our attention to the fibres that are topologically O CP 1 (−1) and the ones corresponding to the black line, the parametrizing space is homeomorphic to S 2 × [0, 1].The remaining fibres are parametrized by B 3 (1), open unit ball of R 3 .As we removed the zero section of S / − (S 4 ), it is clear that we can glue these partial parametrizations together to obtain B 3 (2).Now, β gives a circle action on B 3 (2) that vanishes on its boundary.We conclude that the parametrizing space B of the smooth Cayley fibration is S 4 .Indeed, this is essentially the same way to describe S 4 as we did in Subsection 5.1.
The smoothness of the fibres (the asymptotic analysis).In this subsection, we study the smoothness of the fibres.Observe that this property is obviously satisfied as long as they are contained in the chart U. Hence, the Cayleys of topology S 3 × R are smooth, and we only need to check the remaining ones in the points where they meet the zero section, i.e., when the SU(2) group action degenerates.To this purpose, we carry out an asymptotic analysis.Let β 0 , v 0 , δ 0 and F 0 be the constants determining a Cayley fibre N .By the explicit formula for F , we see that N is given by: We first check the smoothness of the fibres that meet the zero section (u = 0) at some α 0 ∈ (0, π/2), i.e., the ones of topology O CP 1 (−1).For this purpose, if we expand near α − 0 and we obtain the linear approximation of N at that point.Explicitly, this is the SU(2)-invariant 4-dimensional submanifold Σ characterized by the equation and where v, δ, β are constantly equal to v 0 , δ 0 , β 0 .Now, we want to study the asymptotic behaviour of the metric g c when restricted to Σ, and then, we let α tends to α 0 from the left.To do so, it is convenient to compute the following identities using the definition of u := st and v := s/t: (5.24) The metric g c , in the coframe {σ 1 , σ 2 , σ 3 , dα, dβ, du, dv, dδ}, then can be rewritten as: (5.25) where we used (5.24) and Lemma 5.7.Now, if we restrict (5.25) to Σ, and we let α tend to α 0 from the left, we get: As the length of σ 1 is 4π, we deduce that the metric g c extends smoothly to the CP 1 ∼ = S 2 contained in the zero section.This two-dimensional sphere corresponds to the base of the bundle O CP 1 (−1).Finally, we check the smoothness of the fibres meeting the zero section at α 0 = π/2, i.e., the ones with topology R 4 .Expanding for α → π/2 − , we immediately see that the first order is not enough and we need to pass to second order.Explicitly, this is the SU(2)-invariant 4-dimensional submanifold Σ of equation: where A := cv(1 + v 2 ) −1 is the constant depending on c, v determined by the expansion.As above, the remaining parameters v, δ, β are constantly equal to v 0 , δ 0 , β 0 .If we restrict g c as defined in (5.25) to Σ, and we let α tend to π/2, then, we obtain: , where we also used the expansion of cos α around π/2 and the explicit value of A. We conclude that N is not smooth when it meets the zero section, and it develops an asymptotically conical singularity at that point.
The main theorems.We collect all these results in the following theorems.Observe that we are using the notion of Cayley fibration given in Definition 2.8.Theorem 5.16 (Generic case).Let (M, Φ c ) be the Bryant-Salamon manifold constructed over the round sphere S 4 for some c > 0, and let SU(2) act on M as in Subsection 5.3.Then, M admits an SU(2)-invariant Cayley fibration parametrized by B ∼ = S 4 .The fibres are topologically O CP 1 (−1), S 3 × R and R 4 .Apart from the non-vertical fibres of topology R 4 , all the others are smooth.The singular fibres of the Cayley fibration have a conically singular point and are parametrized by (B • ) c ∼ = S 2 × S 1 (β, δ, v in our description).Moreover, at each point of the zero section S 4 ⊂ S / − (S 4 ), infinitely many Cayley fibres intersect.
Theorem 5.17 (Conical case).Let (M 0 , Φ 0 ) be the conical Bryant-Salamon manifold constructed over the round sphere S 4 , and let SU(2) act on M 0 as in Subsection 5.3.Then, M 0 admits an SU(2)invariant Cayley fibration parametrized by B ∼ = S 4 .The fibres are topologically S 3 × R and are all smooth.Moreover, as these do not intersect, the SU(2)-invariant Cayley fibration is a fibration in the usual differential geometric sense with fibres Cayley submanifolds.
Remark 5.18.It is interesting to observe that, in the generic case, the family of singular R 4 s separates the fibres of topology S 3 × R from the ones of topology O CP 1 (−1).
Remark 5.19.Similarly to [KLo21, Subsection 5.11.1], one can blow-up at the north pole and argue that in the limit the Cayley fibration splits into the product of a line R and of an SU(2)-invariant coassociative fibration on R 7 .By the uniqueness of the SU(2)-invariant coassociative fibrations of R 7 , we deduce that the latter is the Harvey and Lawson coassociative fibration [HL82, Section IV.3] up to a reparametrization.
Remark 5.20.From the computations that we have carried out, it is easy to give an explicit formula for the multi-moment map ν c associated to this action.Indeed, this is: Obviously, the range of ν c is the whole R.Under the usual transformation u = st and v = s/t, the multi-moment map becomes: We draw the level sets of ν c in Figure 4.The black lines correspond to the level set relative to zero, the red lines correspond to negative values, while the blue lines correspond to the positive ones.
Differently from the conical case, the 0-level set of ν c for c > 0 does not coincide with the locus of u-minimum of each level set of F .Moreover, for every c ≥ 0, it does not even coincide with the set of SU(2)-orbits of minimum volume in each fibre.
Asymptotic geometry.Inspecting the geometry of the Cayley fibration (see Figure 1), we deduce that there are two asymptotic behaviours for the fibres: one for α ∼ 0 and one for α ∼ π/2.In both cases, as u → ∞, the tangent space of the Cayley fibre N tends to be spanned by ∂ u , ∂ 1 , ∂ 2 , ∂ 3 .We can use the formula for the metric (5.25) to obtain, for α ∼ 0: where, in both cases, When α ∼ π/2, the link S 3 is endowed with the round metric, while, when α ∼ 0, the round sphere is squashed by a factor 1/5.
Remark 5.21.Observe that 1/5 is also the squashing factor on the round metric of S 7 that makes the space homogeneous, non-round and Einstein.It is well-known that there are no other metrics satisfying these properties [Zil82].
6.The Cayley fibration invariant under the lift of the Sp(1) × Id 1 action on S 4 Let M := S / − (S 4 ) and M 0 := R + × S 7 be endowed with the torsion-free Spin(7)-structures Φ c constructed by Bryant and Salamon that we described in Section 3. On each Spin(7) manifold, we construct the Cayley Fibration which is invariant under the lift to M (or M 0 ) of the standard (left multiplication) Sp(1) × Id 1 action on S 4 ⊂ H ⊕ R.
Remark 6.1.The exact same computations will work for the Sp(1) × Id 1 action given by right multiplication of the quaternionic conjugate.In this case, the role of the north and of the south pole will be interchanged.
6.1.The choice of coframe on S 4 .As in Section 5, we choose an adapted orthonormal coframe on S 4 which is compatible with the symmetries we will impose.
Consider R 5 as the sum of a 4-dimensional space P ∼ = H and its orthogonal complement P ⊥ ∼ = R.With respect to this splitting, we can write the 4-dimensional unit sphere in the following fashion: Now, for all (x, y) ∈ S 4 there exists a unique α ∈ [−π/2, π/2] such that x = cos αu, y = sin α, for some u ∈ S 3 .Note that u is uniquely determined when α = ±π/2.Essentially, we are writing S 4 as a 1-parameter family of S 3 s that are collapsing to a point on each end of the parametrization.
Let {∂ 1 , ∂ 2 , ∂ 3 } be the standard left-invariant orthonormal frame on S 3 ∼ = Sp(1).Considering this frame in the description of S 4 above, we deduce that is an oriented orthonormal frame of S 4 \ {α = ±π/2}.The dual coframe is: where {σ i } 3 i=1 is the dual coframe of {∂ i } 3 i=1 in S 3 , which is well-known to satisfy: We deduce that the round metric on the unit sphere S 4 can be written as: and the volume form is: 6.2.The horizontal and the vertical space.Exactly as in Subsection 5.2 we can compute the connection 1-forms ρ i for i = 1, 2, 3 with respect to the coframe we have constructed.Indeed, a straightforward computation involving (3.1), (6.1) and (6.2) implies that ρ i = lσ i for all i = 1, 2, 3, where l := sin α − 1 2 .
Hence, we can deduce from (3.2) that the vertical 1-forms in these coordinates are: 6.3.The SU(2) action.Given the splitting of R 5 into P ∼ = H and its orthogonal complement P ⊥ , we can consider SU(2) ∼ = Sp(1) acting via left multiplication on P and trivially on P ⊥ .Equivalently, we are considering Sp(1) ∼ = Sp(P ) × Id P ⊥ ⊂ SO(5).Being a subgroup of SO(5), the action descends to the unit sphere S 4 .We first consider α = −π/2, where we trivialize S 4 \{south pole} using homogeneous quaternionic coordinates on HP 1 ∼ = S 4 .In this chart, diffeomorphic to H, the action is given by standard left multiplication.
We extend the action on S 4 to the tangent bundle of S 4 via the differential.In this trivialization, H × H, the action is given by left-multiplication on both factors.Hence, if we pick the trivialization of P SO(4) induced by {1, i, j, k}, the action of p ∈ Sp(1) maps the element (x, Id SO(4) ) ∈ H × SO(4) to (p • x, p), where .
As in Section 5, this passes to the quotient space: S / − (S 4 ), and, in the induced trivialization, H × H, the action of Sp(1) is only given by left multiplication on the first factor by definition of µ − .
A similar argument works for the other chart of HP 1 .However, the left multiplication becomes right multiplication of the conjugate, and the lift of the new p is (Id Sp(1) , p).It follows that Sp(1) acts on the fibre over the south pole as it acts on H.
In particular, we proved the following lemma.
Proof.It is straightforward from the definition of dual frame and (6.3).6.5.The Cayley condition.Analogously to the case carried out in Section 5, the generic orbits of the considered SU(2) action are 3-dimensional (see Lemma 6.2).Hence, it is sensible to look for invariant Cayley submanifolds.To this purpose, we assume that the submanifold N consists of a 1-parameter family of 3-dimensional SU(2)-orbits in M .In particular, the coordinates that do not describe the orbits, i.e. a 0 , a 1 , a 2 , a 3 and α, need to be functions of a parameter τ .This means that we can write: The tangent space is spanned by {∂ 1 , ∂ 2 , ∂ 3 , α∂ α + 3 i=0 ȧi ∂ a i }, where the dots denote the derivatives with respect to τ .The condition under which N is Cayley becomes a system of ODEs.Theorem 6.5.Let N be an SU(2)-invariant submanifold as described at the beginning of this subsection.Then, N is Cayley in the chart U if and only if the following system of ODEs is satisfied: Proof.We first write the tangent space of N , which is spanned by {∂ 1 , ∂ 2 , ∂ 3 , α∂ α + 3 i=0 ȧi ∂ a i }, in terms of the frame {e α , e 1 , e 2 , e 3 , e ξ 0 , e ξ 1 , e ξ 2 , e ξ 3 }.This can be easily done using Lemma 6.4.Through a long computation analogous to the one carried out in Appendix A, we can apply Proposition 2.6 to this case, and we obtain the system of ODEs.Remark 6.6.It is interesting to point out that, exactly as in the SO(3) × Id 2 case (see Lemma A.4), the projection π 7 of Proposition 2.6 will just be the identity in the proof of Theorem 6.5.6.6.The Cayley fibration.In the previous section we found the condition that makes N , SU(2)invariant submanifold, Cayley.This consists of a system of ODEs, which will characterize the desired Cayley fibration.
Harvey and Lawson local existence and uniqueness theorem implies that any SU(2)-invariant Cayley can meet the zero section only when α = ±π/2, i.e. outside of U. Otherwise, the zero section of S / − (S 4 ), which is Cayley, would intersect such an N in a 3-dimensional submanifold, contradicting Harvey and Lawson theorem.It follows that the initial value of one of the a i s is different from zero.We take a 0 (0) = 0, as the other cases will follow similarly.Now, it is straightforward to notice that: solves the first 3 equations of the system given in Theorem 6.5.Moreover, it also reduces the remaining equations to the ODE: cos α(−f cos 2 α + 3l 2 gr 2 ) ȧ0 − l(l 2 gr 2 − 3f cos 2 α)a 0 α = 0, where, as usual, r 2 = a 2 0 + a 2 1 + a 2 2 + a 2 3 , l = (sin α − 1)/2, f = 5(c + r 2 ) 3/5 and g = 4(c + r 2 ) −2/5 .As (6.7) implies that a 0 = p −1 r, where p is the positive real number satisfying p 2 = 1 + 3 i=1 (a i (0)/a 0 (0)) 2 , we can rewrite the previous ODE as: cos α(−f cos 2 α + 3l 2 gr 2 ) ṙ − l(l 2 gr 2 − 3f cos 2 α)r α = 0. (6.8) Remark 6.7.It is easy to verify that (6.8) is not in exact form.Hence, it cannot be easily integrated.It is a non-trivial open task to verify whether, possibly up to change of coordinates, (6.8) can be integrated in closed form.
The conical version.We consider the easier conical case first.From a topological point of view, it is obvious that the red and green Cayleys of Figure 6 (B) are homeomorphic to S 3 × R. As the the group action becomes trivial on α = π/2, the topology of the fibres in blue cannot be recovered from the picture.However, it will be clear from the asymptotic analysis that these are smooth topological R 4 s.As a consequence, we have constructed a Cayley fibration on the chart U ∩ M 0 , which extends to the whole M 0 by continuity (i.e.we complete the Cayleys in blue and we add the whole π 0 -fibre at α = −π/2).On M 0 the Cayley fibration remains a fibration in the classical sense.A reasoning similar to the one of Section 5 shows that the parametrizing space B of the Cayley fibration is R 4 .
The smooth version.Now, we deal with the generic case c > 0. As above, the topology of the red Cayleys of Figure 6 (A) is S 3 × R; the blue ones have topology R 4 .In the latter, we use the same asymptotic analysis argument of the conical case.Finally, the submanifolds in green are smooth topological R 4 s.As usual, we extend the Cayley fibration on U to the whole M by continuity (i.e.we add the whole π c -fibre over α = −π/2, we complete the Cayleys in blue and green, and we add the zero section S 4 ).Observe that the zero section, the π c -fibre over α = −π/2 and the green Cayleys all intersect in a point p.It follows that the M given in Definition 2.8 is equal to M \ {p}.Once again, a reasoning similar to the one of Section 5 shows that the parametrizing space B of the Cayley fibration is S 4 .
Consider the two-form given in Proposition 2.6 that projects to η through π 7 .The summands of such two form can be computed through a direct computation involving the terms obtained in Lemma A.1 and Lemma A.2.
Definition 2.8.Let (M, Φ) be a Spin(7) manifold.M admits a Cayley fibration if there exists a family of Cayley submanifolds N b (possibly singular) parametrized by a 4-dimensional space B satisfying the following properties: • M is covered by the family {N b } b∈B ; • there exists an open dense set B • ⊂ B such that N b is smooth for all b ∈ B • ; • there exists an open dense set M ⊂ M and a smooth fibration π : M → B with fibre N b for all b ∈ B.

Remark 4. 2 .
As in [KLo21, Section 4.4], this fibration becomes the trivial Cayley fibration of R 8 = R 4 ⊕ R 4 when we blow-up at any point of the zero section. 5.The Cayley fibration invariant under the lift of the SO(3) × Id 2 action on S 4

( a )Figure 1 .
Figure 1.Level sets of F in the generic and in the conical case

Figure 4 .
Figure 4. Level sets of the multi-moment map in the generic and conical case