The Role of Spin(9) in Octonionic Geometry

Starting from the 2001 Thomas Friedrich's work on Spin(9), we review some interactions between Spin(9) and geometries related to octonions. Several topics are discussed in this respect: explicit descriptions of the Spin(9) canonical 8-form and its analogies with quaternionic geometry as well as the role of Spin(9) both in the classical problems of vector fields on spheres and in the geometry of the octonionic Hopf fibration. Next, we deal with locally conformally parallel Spin(9) manifolds in the framework of intrinsic torsion. Finally, we discuss applications of Clifford systems and Clifford structures to Cayley-Rosenfeld planes and to three series of Grassmannians.


Introduction
One of the oldest evidences of interest for the Spin(9) group in geometry goes back to the 1943 Annals of Mathematics paper by D. Montgomery and H. Samelson [MS43], which classifies compact Lie groups that act transitively and effectively on spheres, and gives the following list: SO(n), U(n), SU(n), Sp(n), Sp(n) · U(1), Sp(n) · Sp(1), G 2 , Spin(7), Spin(9).
In particular, Spin(9) acts transitively on the sphere S 15 through its Spin representation, and the stabilizer of the action is a subgroup Spin(7).
In the following decade, the above groups, with the only exception of Sp(n) · U(1), appeared in the celebrated M. Berger theorem [Ber55] as the list of the possible holonomy groups of irreducible, simply connected, and non symmetric Riemannian manifolds. Next, in the decade after that, D. Alekseevsky [Ale68] proved that Spin(9) is the Riemannian holonomy of only symmetric spaces, namely of the Cayley projective plane and its non-compact dual. Accordingly, Spin(9) started to be omitted in the Berger theorem statement. Much later, a geometric proof of Berger theorem was given by C. Olmos [Olm05], using submanifold geometry of orbits and still referring to possible transitive actions on spheres.
Moreover, in the last decades of the twentieth century, compact examples have been shown to exist for almost all classes of Riemannian manifolds related to the other holonomy groups in the Berger list. References for this are the books by S. Salamon and D. Joyce [Sal89;Joy00;Joy07]. For these reasons, around the year 2000, the best known feature of Spin(9) seemed to be that it was a group that had been removed from an interesting list.
Coming into the new millennium, since its very beginning, new interest in dealing with different aspects of octonionic geometry appeared, and new features of structures and weakened holonomies related to Spin(9) were pointed out. Among the references, there is, notably, the J. Baez extensive Bulletin AMS paper on octonions [Bae02] as well as the not less extensive discussions on his webpage [Bae12]. Next, and from a more specific point of view, there is the Thomas Friedrich paper on "weak Spin(9)-structures" [Fri01], which proposes a way of dealing with a Spin(9) structure, and this was later recognized by A. Moroianu and U. Semmelmann [MS11] to fit in the broader context of Clifford structures. Also, the M. Atiyah and J. Berndt paper in Surveys in Differential Geometry [AB03] shows interesting connections with classical algebraic geometry. Coming to very recent contributions, it is worth mentioning the work by N. Hitchin [Hit18] based on a talk for R. Penrose's 80th birthday, which deals with Spin(9) in relation to further groups of interest in octonionic geometry.
The aim of the present article is to give a survey of our recent work on Spin(9) and octonionic geometry, in part also with L. Ornea and V. Vuletescu, and mostly contained in the references [PP12; PP13; Orn+13; PP15; PPV16; Pic17; Pic18].
Our initial motivation was to give a construction, as simple as possible, of the canonical octonionic 8-form Φ Spin(9) that had been defined independently through different integrals by M. Berger [Ber72] and by R. Brown and A. Gray [BG72]. Our construction of Φ Spin(9) uses the already mentioned definition of a Spin(9)-structure proposed by Thomas Friedrich and has a strong analogy with the construction of a Sp(2) · Sp(1)-structure in dimension 8 (see Section 3 as well as [PP12]). By developing our construction of Φ Spin(9) , we realized that some features of the S 15 sphere can be conveniently described through the same approach that we used. The fact that S 15 is the lowest dimensional sphere that admits more than seven global linearly independent tangent vector fields is certainly related to the Friedrich point of view. Namely, by developing a convenient linear algebra, we were able to prove that the full system of maximal linearly independent vector fields on any S n sphere can be written in terms of the unit imaginary elements in C, H, O and the complex structures that Friedrich associates with Spin(9) (see Section 5 and [PP13]). Another feature of S 15 is, of course, that it represents the total space of the octonionic Hopf fibration, whose group of symmetries is Spin(9) ⊂ SO(16). Here, the Friedrich approach to Spin(9) allows to recognize both the non-existence of nowhere zero vertical vector fields and some simple properties of locally conformally parallel Spin(9)-structures (here, see Theorem 6.1, Section 7, and [Orn+13]). We then discuss the broader contexts of Clifford structures and Clifford systems, that allow us to deal with the complex Cayley projective plane, whose geometry and topology can be studied by referring to its projective algebraic model, known as the fourth Severi variety. With similar methods, one can also study the structure and properties of the remaining two Cayley-Rosenfeld projective planes (for all of this, see Sections 8-10, and [PP15;PPV16]). Finally, Clifford structures and Clifford systems can be studied in relation with the exceptional symmetrical spaces of compact type as well as with some real, complex, and quaternionic Grassmannians that carry a geometry very much related to octonions (see Sections 11 and 12, and [Pic17;Pic18]).
During the years of our work, we convinced ourselves that Spin(9) influences not only 16-dimensional Riemannian geometry, but also aspects related to octonions of some lower dimensional and higher dimensional geometry. It is, in fact, our hope that the reader of this survey can share the feeling of the beauty of Spin(9), that seems to have some role in geometry, besides being a group that had been removed from an interesting list.

Preliminaries, Hopf Fibrations, and Friedrich's work
The multiplication involved in the algebra (O) of octonions can be defined from the one in quaternions (H) by the Cayley-Dickson process: if where h ′ 1 , h ′ 2 are the conjugates of h ′ 1 , h ′ 2 ∈ H. As for quaternions, the conjugation x = h 1 − h 2 e is related to the non-commutativity: xx ′ = x ′ x. The associator vanishes whenever two among x, x ′ , x ′′ ∈ O are equal or conjugate. For a survey on octonions and their applications in geometry, topology, and mathematical physics, the excellent article [Bae02] by J. Baez is a basic reference.
The 16-dimensional real vector space O 2 decomposes into its octonionic lines, that intersect each other only at the origin (0, 0) ∈ O 2 . Here, m ∈ S 8 = OP 1 = O ∪ {∞} parametrizes the set of octonionic lines (l), whose volume elements (ν l ∈ Λ 8 l) allow the following canonical 8-form on O 2 = R 16 to be defined: where p l denotes the orthogonal projection O 2 → l.
The definition of Φ Spin(9) through this integral was given by M. Berger [Ber72], and here we chose the proportionality factor in such a way to make integers and with no common factors the coefficients of Φ Spin(9) as exterior 8-form in R 16 . The notation is motivated by the following: Cor92]). The subgroup of GL(16, R) preserving Φ Spin(9) is the image of Spin(9) under its spin representation into R 16 .
Definition 2.2. Let (M, g) be a 16-dimensional oriented Riemannian manifold. A Spin(9) structure on M is the datum of any of the following equivalent alternatives.
In particular, the existence of a Spin(9) structure depends only on the conformal class of the metric g on M .
The projection O 2 − {0} → OP 1 associated with decomposition into the octonionic lines l m , l ∞ is a noncompact version of the octonionic Hopf fibration: that is the unique surviving possibility when passing from quaternions to octonions from the series of quaternionic Hopf fibrations: Recall that the latter enter into Figure 1 that encodes prototypes of several structures of interest in quaternionic geometry. Figure 1. The prototype of foliations related to locally conformally hyperkähler manifolds.
At the center of the diagram, there is the locally conformally hyperkähler Hopf manifold S 4n+3 × S 1 . All the other manifolds are leaf spaces of foliations on them, such as the 3-Sasakian sphere S 4n+3 , the positive Kähler-Einstein twistor space CP 2n+1 , and the positive quaternion Kähler HP n . Most of the foliations carry similar structures on their leaves, for example, one has locally conformally hyperkähler Hopf surfaces of S 3 × S 1 . This prototype diagram is only an example, since, when a compact locally hyperkähler manifold has compact leaves on the four canonically defined vertical foliations, our diagram still makes sense, albeit in the broader orbifold category (cf. [OP97]).
When n = 3, there are also the octonionic Hopf fibrations, as in Figure 2: In the octonionic case, an additional fibration appears.
These have no arrow connecting OP 1 with HP 3 and CP 7 , since the complex and quaternionic Hopf fibrations are not subfibrations of the octonionic one (cf. [LV92] as well as Theorem 6.1 in the following Section 6).
Coming back to Spin(9), as the title of Th. Friedrich's article [Fri01] suggests, there is a scheme for "weak Spin(9) structures" to include some possibilities besides the very restrictive holonomy Spin(9) condition. Although the original A. Gray proposal [Gra71] to look at "weak holonomies" was much later shown by B. Alexandrov [Ale05] not to produce new geometries for the series of groups quoted in the Introduction, one can refer to the symmetries of relevant tensors to understand the possibilities for any G-structure. We briefly recall a unified scheme that one can refer to, following the presentation of Ref. [Agr06].
By definition, a G-structure on an oriented Riemannian manifold (M n ) is a reduction (R ⊂ F (M n )) of the orthonormal frame bundle to the subgroup G ⊂ SO(n). The Levi Civita connection (Z), thought of as a 1-form on F (M n ) with values in the Lie algebra so(n), restricts to a connection on R, decomposing with respect to the Lie algebra splitting so(n) = g ⊕ m, as Here, Z * is a connection in the principal G-bundle R, and Γ is a 1-form on M n with values in the associated bundle R × G m, called the intrinsic torsion of the G-structure. Of course, the condition Γ = 0 is equivalent to the inclusion Hol ⊂ G for the Riemannian holonomy, and G-structures with Γ = 0 are called non-integrable.
This scheme can be used, in particular, when G is the stabilizer of some tensor η in R n , so that the Gstructure on M defines a global tensor η, and here, Γ = ∇η can be conveniently thought of as a section of the vector bundle: Accordingly, the action of G splits W into irreducible G-components: W = W 1 ⊕ · · · ⊕ W k .
A prototype of such decompositions occurs when G = U(n) ⊂ SO(2n) and yields the four irreducible components of the so-called Gray-Hervella classification when n ≥ 3 [GH80]. It is a fact from the representation theory that there are several further interesting cases that yield four irreducible components. This occurs when G = G 2 ⊂ SO(7), G = Sp(2) · Sp(1) ⊂ SO(8), G = Spin(9) ⊂ SO(16), as computed in Refs. [FG82], [Swa90] (p. 115) and [Fri01], respectively: (2.7) For the mentioned four situations, the last component, W 4 , is the "vectorial type" one, and gives rise to a 1form θ on the manifold. A general theory of G-structures in this last class, W 4 , of typically locally conformally parallel G-structures, was developed in Ref. [AF06]. In Section 7, we revisit W 4 in the G = Spin(9) case, following Ref. [Fri01] as well as our previous work [Orn+13].
These two expressions of Φ Spin(9) have been shown to be proportional according to the following algebraic relation.

The analogy with Sp(2) · Sp(1)
In the previous Section we saw that the matrices I 1 , . . . , I 9 are the starting point for the construction of the canonical 8-form Φ Spin(9) . Of course, I 1 , . . . , I 9 are the octonionic analogues of the classical Pauli matrices which are defined with just the unit imaginary i ∈ C, belonging to U(2). Their compositions (J αβ = I α I β ), for α < β act on H ∼ = C 2 as multiplications on the right by unit quaternions: Similarly, the quaternionic analogues of the Pauli matrices are the 8 × 8 real matrices: where R i , R j , andR k are the multiplication on the right on H by i, j, k. The ten compositions J αβ def = I α I β (α < β) of these latter matrices are a basis of the term sp(2) in the decomposition where ⋆ denotes the Hodge star of what appears before, and On the other hand, the matrices B = B ′ B ′′ B ′′′ B ′′′′ ∈ SO(8) which commute with each of the involutions I 1 , . . . , I 5 are the ones satisfying B ′′ = B ′′′ = 0 and B ′ = B ′′′′ ∈ Sp(1) ⊂ SO(4). Thus, the subgroup preserving each of the I 1 , . . . , I 5 is the diagonal Sp(1) ∆ ⊂ SO(8). Thus, the subgroup of SO(8) preserving the vector bundle E 5 consists of matrices (B) satisfying BI α = I ′ α B, with I 1 , . . . , I 5 and I ′ 1 , . . . , I ′ 5 bases of E 5 related by a SO(5) matrix. This group is thus recognized to be Sp(1) · Sp(2), and the following proposition applies.
Proposition 4.1. Let M 8 be an 8-dimensional oriented Riemannian manifold, and let E 5 be a vector subbundle of End(T M ), locally spanned by self dual anti-commuting involutions (I 1 , . . . , I 5 ) and related, on open sets covering M , by functions giving SO(5) matrices. Then, the datum of such an E 5 is equivalent to an (left) almost quaternion Hermitian structure on M , i.e., to a Sp(2) · Sp(1)-structure on M .
In the above discussions, we looked at the standard U(2) and Sp(1) · Sp(2)-structures on R 4 and R 8 , through the decompositions of the 2-forms , and the orthonormal frames in the so(3) and sp(2) components, respectively. The last components, Λ 2 2 and Λ 2 15 , describe all of the similar structures on the linear spaces R 4 and R 8 . Thus, such decompositions give rise to the SO(4)/U(2) and SO(8)/Sp(1) · Sp(2) spaces-the spaces of all possible structures in the two cases.
To summarize (cf, [GWZ86]), x ′ , when u, x, x ′ ∈ C, H, O (and, in any case, r ∈ R and r 2 + uu = 1) generate the groups U(2), Sp(2) · Sp(1), Spin(9) of symmetries of the Hopf fibrations The corresponding G-structures on the Riemannian manifolds M 4 , M 8 , M 16 can be described through E ⊂ End T M vector subbundles of ranks 3, 5, 9, respectively. Any such E is locally generated by the self-dual involutions I α satisfying I α I β = −I β I α for α = β and related, on open neighborhoods covering M , by functions that give matrices in SO(3), SO(5), and SO(9).

Vector fields on spheres
An application of Spin(9) structures is the possibility of writing a maximal orthonormal system of tangent vector fields on spheres of any dimension. Here, we outline this construction only on some "low-dimensional" cases (in fact. up to the S 511 sphere), referring, for the general case, to the linear algebra formalism developed in Ref. [PP13].
Recall that the identifications R 2n = C n , R 4n = H n and R 8n = O n allow to act on the normal vector field of the unit sphere by the imaginary units of C, H, O, giving 1, 3, and 7 tangent orthonormal vector fields on S 2n−1 , S 4n−1 , and S 8n−1 . These are, in fact, a maximal system of linearly independent vector fields on S m−1 ⊂ R m , provided the (even) dimension (m) of the ambient space is not divisible by 16. The maximal number (σ(m)) of linearly independent vector fields on any S m−1 is well-known to be expressed as where σ(m) + 1 = 2 p + 8q is the Hurwitz-Radon number, referring to the decomposition for further information and references on this classical subject). Table A lists some of the lowest dimensional S m−1 ⊂ R m spheres that admit a maximal number σ(m) > 7 of linearly independent vector fields. Table A. Some S m−1 spheres with more than seven vector fields. The first of them is S 15 ⊂ R 16 , which is acted on by Spin(9) ⊂ SO(16). To write the eight vector fields on S 15 , it is convenient to look at the involutions I 1 , . . . , I 9 and at the eight complex structures (J 1 , . . . , J 8 ) on . , x 8 , y 1 , . . . , y 8 ) the (outward) unit normal vector field of S 15 ⊂ R 16 . Then, the following proposition applies. process. By denoting the canonical basis of S over R by 1, e 1 , . . . , e 15 , one can write a multiplication table (cf. Ref. [PP13]). An example of divisors of the zero in S is given by (e 2 − e 11 )(e 7 + e 14 ) = 0.
The following remark helps in higher dimensions. Consider the S m−1 ⊂ R m sphere, and decompose m as m = (2k + 1)2 p 16 q , where p ∈ {0, 1, 2, 3}. First, observe that a vector field (B) that is tangential to the S 2 p 16 q −1 ⊂ R 2 p 16 q sphere induces a vector field that is tangential to the S (2k+1)2 p 16 q −1 sphere. Thus, assume in what follows that k = 0, i.e., m = 2 p 16 q . Whenever we extend a vector field in this way, we call the vector field given by (5.3) the diagonal extension of B.
If q = 0, that is, if m is not divisible by 16, the vector fields on S m−1 are given by the complex, quaternionic, or octonionic multiplication for p = 1, 2, or 3 respectively, so that the Spin(9) contribution occurs when q ≥ 1, that is, m = 16l, and we can denote the coordinates in R 16l by (s 1 , . . . , s l ), where each s α , for α = 1, . . . , l, belongs to the sedenions (S), and can thus be identified with a pair ((x α , y α )) of octonions.
The unit (outward) normal vector field (N ) of S 16l−1 can be denoted by using the sedenions: Therefore, we can think of N as an element of S l = O 2l = R 16l . Whenever l = 2, 4, or 8, denoted by D, the following automorphism of S l = O 2l applies.
We refer to D as a conjugation, due to its similarity to that in * -algebras. Moreover, it is convenient to use the following formal notations: which allow us to define left multiplications (L) in the sedenionic spaces S 2 , S 4 , and S 8 (like in C, H, and O), as follows.
If l = 2, the left multiplication is whereas, if l = 4, we define and finally, if l = 8, we define (5.10) Note that, in all three cases, l = 2, 4, and 8, and the vector fields L i (N ), . . . , L h (N ) are tangential to S 31 , S 63 , and S 127 , respectively.
We can now write the maximal systems of vector fields on S 31 , S 63 and S 127 as follows.
For S 31 , whose maximal number of tangent vector fields is nine, we obtain eight vector fields by writing the unit normal vector field as N = (s 1 , s 2 ) = (x 1 , y 1 , x 2 , y 2 ) ∈ S 31 ⊂ S 2 , where x 1 , y 1 , x 2 , y 2 ∈ O, and repeating Formula (5.2) for each pair ((x 1 , y 1 ), (x 2 , y 2 )): A ninth orthonormal vector field, completing the maximal system, is found by the formal left multiplication (5.8) and the D automorphism (5.4): (5.12) The S 63 sphere has a maximal number of 11 orthonormal vector fields. The normal vector field is, in this case, given by N = (s 1 , . . . , s 4 ) = (x 1 , y 1 , . . . , x 4 , y 4 ) ∈ S 63 ⊂ S 4 , and eight vector fields arise as J α N , for α = 1, . . . , 8. Three other vector fields are again given by the formal left multiplications (5.9) and the D automorphism in (5.4): and one gets the following tangent vectors that can be easily checked to be orthonormal: To see that each J α N vector is orthogonal to each D(block(J β )N ), for α, β = 1, . . . , 8, look at the matrix representations of R i , . . . , R h and write the octonionic coordinates as x λ = h λ 1 + h λ 2 e, y µ = k µ 1 + k µ 2 e. Then, the scalar product < J α N, D(block(J β )N ) > can be computed with Formula (2.1) to obtain the product of the octonions. For example, recall from Formula (5.18) that To conclude, observe that the real part (ℜ) of the sums of each of the corresponding underlined terms is zero. This is due to the identity (ℜ(hh ′ h ′′ ) = ℜ(h ′ h ′′ h)), that holds for all h, h ′ , h ′′ ∈ H.
More generally, the following proposition applies.
Proposition 5.3. Fix any β, 1 ≤ β ≤ 9, consider the eight complex structures I α I β , with α = β, defined on R 256 = R 16 ⊕ · · · ⊕ R 16 by acting with the corresponding matrices on the listed 16-dimensional components, that is, the diagonal extension of I α I β . Also, consider the further eight complex structures D(block(I α I β )) for α = β, defined by the same matrices that now act on the column matrix of sedenions ((s 1 , . . . s 16 ) T ). Then, The m = 2 · 16 2 dimension, that is, the S 511 sphere, is the lowest dimensional case where the last ingredient of our construction enters. To define the additional vector field here, we need to extend the formal left multiplication defined by Formula (5.8). Consider the decomposition R 2·16 2 = R 16 2 ⊕ R 16 2 and denote the elements in R 16 2 by s 1 , s 2 . Using the notation To make things work, we need to extend not only L i , but also the D conjugation. To this aim, the elements s α ∈ R 16 2 are split into (x α , y α ) where x α , y α ∈ R 16 2 /2 , and the conjugation D 2 is defined on R 16 2 using Formula (5.4): D 2 : ((x 1 , y 1 ), (x 2 , y 2 )) −→ ((x 1 , −y 1 ), (x 2 , −y 2 )).
(5.21) The additional vector field is then D(D 2 (L i N )), and the following theorem applies.

Back to the octonionic Hopf fibration
As we saw in the previous Section, S 15 is the lowest dimensional sphere with more than seven linearly independent vector fields. There are further features that distinguish S 15 among spheres. For example, S 15 is the only sphere that admits three homogeneous Einstein metrics, and it is the only sphere that appears as a regular orbit in three cohomogeneity actions on projective spaces, namely, of SU(8), Sp(4), and Spin(9) on CP 8 , HP 4 , and OP 2 , respectively (see Refs. [Bes87;Kol02]). All of these features can be traced back to the transitive action of Spin(9) on the octonionic Hopf fibration S 15 → S 8 . The following theorem applies.
Theorem 6.1. Any global vector field on S 15 which is tangent to the fibers of the octonionic Hopf fibration S 15 → S 8 has at least one zero.
(6.1) Their span, EN def = < I 1 N, . . . , I 9 N >, is, at any point, a nine-plane in R 16 that is not tangential to the S 15 sphere. Observe that the nine-plane EN is invariant under Spin(9). This is certainly the case for the single vector field N , since Spin(9) ⊂ SO(16). On the other hand, the endomorphisms I α rotate under the Spin(9) action inside their E 9 ⊂ End(R 16 ) vector bundle.
Next, note that EN contains N : where the coefficients λ α are computed from (6.1) in terms of the inner products of vectors, x = (x 1 , . . . , x 8 ), y = (y 1 , . . . , y 8 ) ∈ R 8 , and of the right translations, R i , . . . , R h , as follows: In particular, at points with x = 0, that is, on the octonionic line l ∞ , the I 1 N, . . . , I 9 N vector fields are orthogonal to the S 7 ⊂ l ∞ unit sphere. The latter is the fiber of the Hopf fibration S 15 → S 8 over the north pole ((0, . . . , 0, 1) ∈ S 8 ), and the mentioned orthogonality of this fiber (S 7 ) is immediate from (6.1) for I 1 N, . . . , I 8 N.
Also, at these points, we have I 9 N = N , so I 9 N is orthogonal to the S 7 fiber as well. Now, the invariance of the octonionic Hopf fibration under Spin(9) shows that all its fibers are characterized as being orthogonal to the vector fields I 1 N, . . . , I 9 N in R 16 . Now, assume that X is a vertical vector field of S 15 → S 8 . From the previous characterization, we have the following orthogonality relations in R 16 : for α = 1, . . . , 9, and it follows that I α X, N = 0. However, from the definition of a Spin(9) structure, it can be observed that if α = β, then I α X, I β X = 0. Thus, if X is a nowhere zero vertical vector field, we obtain, in this way, nine pairwise orthogonal vector fields (I 1 X, . . . , I 9 X) that are all tangent to S 15 . However, S 15 is known to admit, at most, eight linearly independent vector fields. Thus, X cannot be vertical and nowhere zero.
One gets, as a consequence, the following alternative proof of a result, established in Ref. [LV92].
Proof. In fact, any S 1 subfibration would give rise to a real line sub-bundle (L ⊂ T vert (S 15 )) of the vertical subbundle of T (S 15 ). This line bundle (L) is necessarily trivial, due to the vanishing of its first Stiefel-Whitney class, w 1 (L) ∈ H 1 (S 15 ; Z 2 ) = 0. It follows that L would admit a nowhere zero section and thus, a global, vertical, nowhere zero vector field.

Locally conformally parallel Spin(9) manifolds
Let G ⊂ SO(d). Recall that a locally conformally parallel G-structure on a manifold M d is the datum of a Riemannian metric (g) on M, a covering U = {U α } α∈A of M, and for each α ∈ A, a metric g α defined on U α which has holonomy contained in G such that the restriction of g to each U α is conformal to g α : Some of the possible cases here are • G = U(n), where we have the locally conformally Kähler metrics; • G = Sp(n) · Sp(1), yielding the locally conformally quaternion Kähler metrics; • G = Spin(9), which is the case we are dealing with.
In any of the cases above, for each overlapping U α ∩ U β , the functions f α , f β differ by a constant: This implies that df α = df β on U α ∩ U β = ∅, hence defining a global, closed 1-form that is usually denoted by θ and called the Lee form. Its metric dual with respect to g is denoted by N as N = θ ♯ and is called the Lee vector field.
The G = U(n) case of locally conformally Kähler metrics has been extensively studied in the last decades (see, for instance, Ref. [DO98]).
As mentioned in the Introduction, the holonomy of Spin(9) is only possible on manifolds that are either flat or locally isometric to OP 2 or to the hyperbolic Cayley plane OH 2 . Weakened holonomy conditions give rise to several classes of Spin(9) structures (cf. Ref. [Fri01] and Section 2). One of these classes is that of vectorial type structures (see Refs. [AF06] and [Fri01] (p. 148)). According to the following Definition and the following Remark, this class fits into the locally conformally parallel scheme.
Definition 7.1. [AF06] A Spin(9) structure is of the vectorial type if Γ lives in P 0 .
Remark 7.2. In Refs. [Fri01;AF06], the class of locally conformally parallel Spin(9) structures has been identified and studied under the name Spin(9) structures of vectorial type. Now, we outline now a proof that, for Spin(9) structures, the vectorial type is equivalent to the locally conformally parallel type. As already mentioned in Section 2, the splitting of the Levi-Civita connection, viewed as a connection in the principal bundle of orthonormal frames on M is where Z * is the connection of Spin(9)-frames in the induced bundle, and Γ is its orthogonal complement. Thus, Γ is a 1-form with values in the orthogonal complement m in the splitting of Lie algebras so(16) = spin(9) ⊕ m, and under the identification Λ 2 84 = m = Λ 3 (E 9 ) (cf. the beginning of Section 3), Γ can be seen as a 1-form with values in Λ 3 (E 9 ). Under the action of Spin(9), the space Λ 1 (M ) ⊗ Λ 3 (E) decomposes as a direct sum of four irreducible components: and looking at all the possible direct sums, this yields 16 types of Spin(9) structure. Component P 0 identifies with Λ 1 (M ), and thus, with the component W 4 in Formula (2.7). Now, let (M 16 , g) be a Riemannian manifold endowed with a Spin(9) structure of the vectorial type. Let Γ be as above, and let Φ be its Spin(9)-invariant 8-form. Now, Γ = 0 implies that the holonomy of M is contained in Spin(9) (cf. Ref. [Fri01] (p. 21)).
From Ref. [AF06] (p. 5), we know that the following relations hold: Let (M,g) be the Riemannian universal cover of (M, g), and letΦ,θ be the lifts of Φ and θ respectively. Then, relations (7.1) also hold forΦ andθ. SinceM is simply connected,θ = df for some f :M → R. Then, by defining g 0 def = e −fg and Φ 0 def = e −4fΦ , we have dΦ 0 = 0, that is, the θ-factor of Φ 0 is zero. Hence, g 0 has holonomy contained in Spin(9), and on the other hand, it is locally conformal to g. Thus, M can be covered by open subsets on which the metric is conformal to a metric with holonomy in Spin(9).
The conformal flatness of metrics with Spin(9) holonomy has the following consequences (cf. Ref. [Orn+13] for the proofs).
Theorem 7.3. Let M 16 be a compact manifold that is equipped with a locally, non globally, conformally parallel Spin(9) metric g. Then, (1) The Riemannian universal covering (M ,g) of M is conformally equivalent to the euclidean R 16 \ {0}, the Riemannian cone over S 15 , and M is locally isometric to S 15 × R up to its homotheties. (1) M is the total space of a fiber bundle M π −→ S 1 r , where π is a Riemannian submersion over a circle of radius r.
(2) The fibers of π are spherical space forms (S 15 /K), where K is a finite subgroup of Spin(9).
(3) The structure group of π is contained in the normalizer of K in Spin(9).

Clifford systems and Clifford structures
The self dual anti-commuting involutions I 1 , . . . , I 9 that define the standard Spin(9)-structure on R 16 are an example of a Clifford system. The definition, formalized in 1981 by D. Ferus, H. Karcher, and H. F. Münzner, in their study of isometric hypersurfaces of spheres [FKM81], is the following.
Definition 8.1. A Clifford system on the Euclidean vector space R N is the datum C m = (P 0 , . . . , P m ) of a (m + 1)-ple of symmetric endomorphisms P α such that P 2 α = Id for all α, P α P β = −P β P α for all α = β.
A Clifford system on R N is said to be irreducible if R N is not a direct sum of two positive dimensional subspaces that are invariant under all the P α .
From the representation theory of Clifford algebras, it is recognized (cf. Refs. [FKM81] (p. 483) and [Hus94] (p. 163)) that R N admits an irreducible Clifford system (C = (P 0 , . . . , P m )) if and only if N = 2δ(m), where δ(m) is given as in the following Table B.
Uniqueness can be discussed as follows. Given, on R N , two Clifford systems (C m = (P 0 , . . . , P m ) and C ′ m = (P ′ 0 , . . . , P ′ m )), they are said to be equivalent if A ∈ O(N ) exists such that P ′ α = A t P α A for all α. Then, for m ≡ 0 mod 4, there is a unique equivalence class of irreducible Clifford systems, and for m ≡ 0 mod 4, there are two, which are classified by the two possible values of the trace tr (P 0 P 1 . . . P m ) = ±2δ(m).  In Ref. [PPV16], we outlined the following inductive construction for the irreducible Clifford systems on real Euclidean vector spaces (R N ), taking, as starting the point, the basic Clifford systems (C 1 , C 2 , C 4 , C 8 ) associated with structures given by U(1), U(2), Sp(2) · Sp(1), and Spin(9). All the cases appearing in Tables B and C make sense in the natural context of Riemannian manifolds. We get the following theorem (see Ref. [PPV16] for details).
Theorem 8.2. (Procedure to write new Clifford systems from old). Let C m = (P 0 , P 1 , . . . , P m ) be the last (or unique) Clifford system in R N . Then, the first (or unique) Clifford system, in R 2N has, respectively, the following first and last endomorphisms: where the blocks are N × N . The remaining matrices are 1, . . . , m).
Here, P 0α are the complex structures given by P 0 P α compositions in the Clifford system C m . When the complex structures (P 0α ) can be viewed as (possibly block-wise) right multiplications by some of the unit quaternions (i, j, k) or unit octonions (i, j, k, e, f, g, h), and if the dimension permits it, further similarly defined endomorphisms (Q β ) can be added by using some others among i, j, k or i, j, k, e, f, g, h. Table C. Clifford systems C m and G-structures on Riemannian manifolds (M N ). The notion of an even Clifford structure, a kind of unifying notion proposed by A Moroianu and U. Semmelmann [MS11], is instead given by the following datum on a Riemannian manifold ((M, g)).
Definition 8.3. An even Clifford structure on (M, g) is a real oriented Euclidean vector bundle (E, h), together with an algebra bundle morphism (ϕ : Cl 0 (E) → End(T M )) which maps Λ 2 E into skew-symmetric endomorphisms.
By definition, a Clifford system always gives rise to an even Clifford structure, but there are some even Clifford structures on manifolds that cannot be constructed, even locally, from Clifford systems. An example of this is given by a Spin(7) structure on any oriented 8-dimensional Riemannian manifold as a consequence of the following observations (cf. Ref. [PP12] for further details).
Proposition 8.4. Let C m = {P 0 , . . . , P m ) be a Clifford system in R n . The compositions J αβ def = P α P β for α < β, and J αβγ def = P α P β P γ for α < β < γ, are linearly independent complex structures on R n .
Proof. It can be easily recognized that J αβ and J αβγ are complex structures. On the other hand, for any α = 0, . . . , m, it can be observed that tr(P * α P α ) = 1, and for α < β, tr(P * α P β ) = 0, so that the P α are orthonormal and symmetric. By a similar argument, tr(J * αβ J αβ ) = 1 and tr(J * αβ J γδ ) = tr(P β P α P γ P δ ) = 0 if γ or δ equals α or β. Also, for α = γ and β = δ, the J * αβ J γδ is the composition of the skew-symmetric J βαγ and the symmetric P δ , and as such, its trace is necessarily zero. Similar arguments show that the J αβγ for α < β < γ are orthonormal.
Corollary 8.5. The Spin(7)-structures on R 8 cannot be defined through Clifford system C 6 .
Proof. For any choice of such a Clifford system, C 6 = (P 0 , . . . P 6 ) in R 8 , the complex structures J αβγ for α < β < γ give rise to 35 linearly independent skew-symmetric endomorphisms, contradicting the decomposition of 2-forms in R 8 under Spin (7): Nevertheless, the right multiplications by i, j, k, e, f, g, h ∈ O span the E 7 ⊂ End − R 8 vector bundle, and this identifies Spin(7) structures among the even Clifford structures.
The following Sections present further examples of such essential Clifford structures, i.e., Clifford structures not coming from Clifford systems.
On Riemannian manifolds ((M, g)), it is natural to consider the following class of even Clifford structures.
Definition 8.6. The even Clifford structure E r on (M, g) is said to be parallel if a metric connection (∇ E ) exists on E such that ϕ is connection preserving, i.e., ϕ(∇ E X σ) = ∇ g X ϕ(σ) for every tangent vector X ∈ T M and section σ of Cl 0 E, and where ∇ g is the Levi-Civita connection. Table D summarizes the non-flat, parallel, even Clifford structure, as classified in Ref. [MS11]. The noncompact duals of the appearing symmetric spaces have to be added. A good part of the listed manifolds appear in the following sections. This section deals with E III = E 6 /Spin(10) · U(1) ∼ = V 78 16 ⊂ CP 26 , the second, after the Cayley projective plane (F II), of the exceptional symmetric spaces appearing in Table D. A remarkable feature of E III, one of the two exceptional Hermitian symmetric spaces of compact type, is its model as a smooth projective algebraic variety of complex dimension 16 and degree 78, the so-called fourth Severi variety V 78 16 ⊂ CP 26 . This name was proposed by F. Zak [Zak85], who classified the smooth projective algebraic varieties (V n ) in a CP N that, in spite of their critical dimension (n = 2 3 (N − 2)), are unable to fill CP N through their chords.
On the other hand, E III admits a construction that is very similar to the one of the Cayley projective plane (F II). One can, in fact, look at the complex octonionic Hermitian matrices which are acted on by E 6 with three orbits on CP 26 . The closed one consists of Z matrices of rank one, and, as such, can be thought as (virtual) "projectors on complex octonionic lines in (C ⊗ O) 3 "; thus, they are points of the complex projective Cayley plane E III = E 6 /Spin(10) · U(1) ⊂ CP 26 . The projective algebraic geometry of E III ⊂ CP 26 was studied in detail in Ref. [IM05]. Similarly to Corollary 8.5, we have the following proposition.
Proposition 9.1. The complex space C 16 does not admit any family of ten endomorphisms (P 0 , . . . , P 9 ) that satisfies the properties of a Clifford system and is compatible with respect to the standard Hermitian scalar product g.
Proof. The family P 0 , . . . , P 9 would define (after multiplying each of them by i) a representation of the complex Clifford algebra Cl 10 ∼ = C(32) (the order 32 complex matrix algebra) on the C 16 vector space.
Note, however, that the Euclidean space R 32 admits the Clifford system C 9 (cf. Table B). The parallel, even Clifford structure on E III can be defined through the following one, here described on the C 16 model space.
Moreover, the following theorem applies.
(ii) By looking at E III as the fourth Severi variety (V 78 16 ⊂ CP 26 ), the de Rham dual of the basis represented in H 8 (E III; Z) by the forms ( 1 (2π) 4 Φ Spin(10) , 1 (2π) 4 ω 4 ) is given by the pair of algebraic cycles where CP 4 , (CP 4 ) ′ are maximal linear subspaces that belong to the two different families ruling a totally geodesic, non-singular, quadric Q 8 contained in V 78 16 .

Cayley-Rosenfeld planes
Besides the real and the complex Cayley projective planes F II and E III, there are two further exceptional symmetric spaces of compact type, that are usually referred to as the Cayley-Rosenfeld projective planes, namely, the "projective plane over the quaternionic octonions": E VI = E 7 /Spin(12) · Sp(1) = (H ⊗ O)P 2 , and the "projective plane over the octonionic octonions":

By referring to the inclusions
it can be observed that the projective geometry of these four projective planes, which is notably present on the first two steps, becomes weaker at the third and fourth Cayley-Rosenfeld projective planes (cf. Ref. [Bae02]). However, the dimensions of these four exceptional symmetric spaces are coherent with this terminology. According to Table D, these four spaces have the highest possible ranks for non-flat, parallel, even Clifford structures.
The following Table E summarizes the even Clifford structures on the four Cayley-Rosenfeld planes. Only the one on F II is given by a Clifford system; the other three are essential. Concerning the cohomology generators, the one in dimension 8 can also be constructed for E VI and E VIII via the fourth coefficient (τ 4 (ψ)) of the matrices (ψ) of Kähler 2-forms that are associated with the even Clifford structures that we are now listing. Remark 10.1. The matrices of the Kähler 2-forms that are associated with the even Clifford structures go through spin(9) ⊂ spin(10) ⊂ spin(12) ⊂ spin(16), and the last Lie algebra decomposes as spin(16) = so(16) = spin(9) ⊕ Λ 2 84 .

Exceptional symmetric spaces
A good number of symmetric spaces that appear in Table D  and E VII = E 7 E 6 · U(1) , are Kähler and therefore, are equipped with a non-flat, parallel, even Clifford structure of rank r = 2. Next, the five Wolf spaces (1) , which are examples of positive quaternion Kähler manifolds, carry a rank of r = 3 in a non-flat, parallel, Clifford structure. Thus, seven of the twelve exceptional, compact type, Riemannian symmetric spaces of are either Kähler or quaternion Kähler. Accordingly, one of their de Rham cohomology generators is represented by a Kähler or quaternion Kähler form, and any further cohomology generators can be viewed as primitive in the sense of the Lefschetz decomposition.
As seen in the previous sections, the four Cayley-Rosenfeld projective planes, E III, E VI, E VIII, F II carry a similar structure with r = 10, 12, 16, 9. Thus, among the exceptional symmetric spaces of compact type, there are two spaces that admit two distinct even Clifford structures, namely, the Hermitian symmetric E III has even Clifford structures of rank 2 and of rank 10, and the quaternion Kähler E VI has even Clifford structures of rank 3 and rank of 12. For simplicity, we call octonionic Kähler the parallel even Clifford structure defined by the vector bundles E 10 , E 12 , E 16 , E 9 on the Cayley-Rosenfeld projective planes (E III, E VI, E VIII, F II). In conclusion, and with the exceptions of nine of the twelve exceptional Riemannian symmetric spaces of compact type admit at least one parallel, even Clifford structure. Any of such structures gives rise to a canonical differential form: the Kähler 2-form ω for the complex Kähler one, the quaternion Kähler 4-form Ω for the five Wolf space, and a canonical octonionic Kähler 8-form Ψ for the four Cayley-Rosenfeld projective planes. Their classes are always one of the cohomology generators, and Table F collects some informations on the exceptional symmetric spaces of compact type. For each of them, the real dimension, the existence of torsion in the integral cohomology, the Kähler or quaternion Kähler or octonionic Kähler (K/qK/oK) property, the Euler characteristic χ, and the Poincaré polynomial (up to mid dimension) are listed.
of the nine exceptional Riemannian symmetric spaces that admit an even parallel Clifford structure. Here, the meaning of "primitive" varies depending on the considered K/qK/oK structure. Thus, the Hermitian symmetric spaces, E III and E VII, are simply polynomials with coefficients the primitive Betti numbers, where L ω is the Lefschetz operator which multiplies the cohomology classes with the complex Kähler form ω, and n is the complex dimension.
In the positive quaternion Kähler setting, the vanishing of odd Betti numbers and the injectivity of the Lefschetz operator L Ω : H 2k−4 → H 2k , k ≤ n, now occur with Ω being the quaternion 4-form and n being the quaternionic dimension. A remarkable aspect of the primitive Betti numbers for positive quaternion Kähler manifolds is their coincidence with the ordinary Betti numbers of the associated Konishi bundle-the 3-Sasakian manifold fibering over it (cf. [GS96] (p. 56)).
Thus, the following theorem applies.

Poin
Gr 2 (H 4 ) = 1 + t 4 + 2t 8 + t 12 + t 16 . Examples 12.6. Finally, we mention some higher dimensional examples. First, we address the 32-dimensional Wolf space Gr 8 (R 12 ), that has three non-flat, even Clifford structures: two of rank 3, corresponding to the two quaternion Kähler structures (in correspondence with two different ways to define hypercomplex structures on the planes on any Gr 4 (R n+4 )), and one of rank 8, described in this section. Indeed, Gr 8 (R 12 ) can be looked at as the "quaternion octonionic" projective line ((H ⊗ O)P 1 ) that is a total geodesic sub-manifold of the exceptional symmetric space (H ⊗ O)P 2 ∼ = E VI, cf. [Esc12]. Its Poincaré polynomial,