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Clifford systems, Clifford structures, and their canonical differential forms

Clifford systems, Clifford structures, and their canonical differential forms 2 8 A comparison among different constructions in ℍ ≅ ℝ of the quaternionic 4-form Φ and of the Cayley calibration Φ shows that one can start for them from the Sp(2)Sp(1) Spin(7) same collections of “Kähler 2-forms”, entering both in quaternion Kähler and in Spin(7) geometry. This comparison relates with the notions of even Clifford structure and of Clif- ford system. Going to dimension 16, similar constructions allow to write explicit formulas in ℝ for the canonical 4-forms Φ and Φ , associated with Clifford systems Spin(8) Spin(7)U(1) related with the subgroups Spin(8) and Spin(7)U(1) of . We characterize the cali- SO(16) brated 4-planes of the 4-forms Φ and Φ , extending in two different ways the Spin(8) Spin(7)U(1) notion of Cayley 4-plane to dimension 16. Keywords Octonions · Clifford system · Clifford structure · Calibration · Canonical form Mathematics Subject Classification Primary 53C26 · 53C27 · 53C38 1 Introduction In 1989 R. Bryant and R. Harvey defined the following calibration, of interest in hyper - kähler geometry [6]: 1 1 1 2 2 2 4 n Φ =−  −  +  ∈Λ ℍ . R R R i j k 2 2 2 Communicated by Vicente Cortés. * Paolo Piccinni piccinni@mat.uniroma1.it Kai Brynne M. Boydon kbboydon@math.upd.edu.ph Institute of Mathematics, University of the Philippines Diliman, Quezon City, Philippines Dipartimento di Matematica, Sapienza Università di Roma, Piazzale Aldo Moro 2, 00185 Roma, Italy 1 3 Vol.:(0123456789) 102 K. B. M. Boydon, P. Piccinni In this definition, ( , , ) are the Kähler 2-forms of the hypercomplex structure R R R i j k (R , R , R ) , defined by multiplications on the right by unit quaternions (i,  j, k) on the space i j k 4n n ℝ ≅ ℍ . When n = 2 , the Bryant-Harvey calibration Φ relates with Spin(7) geometry. This is easily recognized by using the map L ∶ ℍ → 𝕆 , L(h , h )= h +(kh k)e ∈ 𝕆 , 1 2 1 2 from pairs of quaternions to octonions, that yields the identity L Φ =Φ . Spin(7) K (1.1) 4 8 Here Φ ∈Λ ℝ is the Spin(7) 4-form, or Cayley calibration, studied since the R. Har- Spin(7) vey and H. B. Lawson’s foundational paper [11], and defined through the scalar product and the double cross product of ℝ ≅ 𝕆 : Φ (x, y, z, w)= < x , y × z × w > = < x , y(zw ̄ ) >, Spin(7) assuming here orthogonal y, z, w ∈ . The present paper collects some of the results in the first author Ph.D. thesis [3], inspired from viewing formula (1.1) as a way of constructing the Cayley calibration Φ Spin(7) through the 2-forms  , , . As well known, by summing the squares of the latter R R R i j k 2-forms one gets another remarkable calibration, namely the quaternionic right 4-form Ω . Thus  , , , somehow building blocks for quaternionic geometry, enter also in R R R i j k Spin(7) geometry. A first result is the following Theorem  1.1, a kind of “other way around” of formula (1.1). To state it, recall that the Cayley calibration Φ can also be constructed as sum of Spin(7) squares of “Kähler 2-forms” associated with complex structures on ℝ ≅ 𝕆 , defined by the unit octonions. In fact, cf. [19, Prop.10]: 1 1 2 2 2 2 2 2 2 Φ =− ( +  +⋯ +  )= ( +  +  +⋯ +  ). (1.2) Spin(7) i j h ij ik ik gh 6 6 ,  ,… ,  are the Kähler 2-forms associated with the 7 complex structures Here i j h R , R ,… , R on ℝ ≅ 𝕆 , the right multiplications by the unit octonions i,  j,  k,  e,  f,  g,  h, i j h ,  ,… ,  are the Kähler 2-forms associated with the 21 complex structures and ij ik gh R = R ◦R , R = R ◦R ,… , R = R ◦R , compositions of them. ij i j ik i k gh g h 4 2 Theorem 1.1 The right quaternionic 4-form Ω ∈Λ ℍ can be obtained from the Kähler forms  ,  ,… ,  associated with the complex structures R , R ,… , R as: i j h i j h 2 2 2 2 2 2 2 Ω = 2[ +  +  −  −  −  −  ]. i j k e f g h Moreover, by selecting any five out of the seven (J = R , J = R , … , J = R ) and by look- 1 i 2 j 7 h ing at the matrix  =( )∈ (5) of Kähler 2-forms of their compositions J = J ◦J , one can get the left quaternionic 4-form Ω as Ω =− 𝜁 , 𝛼𝛽 𝛼<𝛽 up to a permutation or change of signs of some coordinates in ℝ . 1 3 Clifford systems, Clifford structures, and their canonical… 103 On the same direction as in Bryant-Harvey’s formula (1.1), a similar result is the follow- ing (cf. Sect. 4 for more details): 4 8 Theorem  1.2 The Cayley calibration Φ ∈Λ ℝ can be obtained from the Kähler Spin(7) L L L L L 2-forms  (1 ≤ 𝛼 <𝛽 ≤ 5) associated to complex structures I = I ◦I , where I ,… I 1 5 are anti-commuting self-dual involutions in ℝ . Namely: 2 2 2 2 2 2 2 2 2 2 Φ =  +  +  +  −  −  −  −  −  −  , Spin(7) 12 13 24 34 23 14 15 25 35 45 and on the other hand one can get the right quaternion Kähler as: 2 2 2 2 2 2 2 2 2 2 Ω =−  +  +  +  +  +  +  +  +  +  . 12 13 24 34 23 14 15 25 35 45 Moving to dimension 16 and in Sect. 6, we consider two exterior 4-forms 4 16 Φ , Φ ∈Λ ℝ , Spin(8) Spin(7)U(1) canonically associated with subgroups Spin(8), Spin(7)U(1) ⊂ SO(16) , and we write their explicit expressions in the 16 coordinates. We will see in the next two statements which 4-planes of ℝ are calibrated by Φ and by Φ . Spin(8) Spin(7)U(1) The 4-forms Φ , Φ and the respective calibrated 4-planes can be compared Spin(8) Spin(7)U(1) with other calibrations in ℝ , in particular with the previously mentioned Bryant-Harvey 4-form Φ . It is thus appropriate to remind the main theorem in [6, Theorem 2.27], namely n 4n that, in any ℍ ≅ ℝ , the Bryant-Harvey 4-form Φ calibrates the Cayley 4-planes that are 2 n contained in a quaternionic 2-dimensional vector subspace W ⊂ ℍ . Here we prove: 16 2 Theorem 1.3 The oriented 4-planes of ℝ ≅ 𝕆 calibrated by the 4-form Φ are the Spin(8) transversal Cayley 4-planes, i.e. the 4-planes P such that both projections (P) ,  (P) on 2 � the two summands in  =  ⊕  are two dimensional and both invariant by a same com- plex structure u ∈ S ⊂ Im. Also, by recalling that  decomposes in the union of octonionic lines = {(x, mx), x ∈ , m ∈  ∪ ∞}, meeting pairwise only at (0, 0)∈  , we can state: Theorem  1.4 The oriented 4-planes of ℝ calibrated by the 4-form Φ are the Spin(7)U(1) ones that are invariant under a complex structure u ∈ S ⊂ Im and that are contained in an octonionic line  ⊂  ⊕  , where only m ∈ ℝ and m =∞ are allowed. Thus they are “Cayley 4-planes”, contained in the oriented 8-planes that are the mentioned octonionic lines with m ∈ ℝ ∪∞. 2 Preliminaries The multiplication in the algebra  of octonions can be defined from the one in quaternions � � � ℍ through the Cayley-Dickson process: if x = h + h e, x = h + h e ∈  , then 1 2 1 2 1 3 104 K. B. M. Boydon, P. Piccinni � � � � � ̄ ̄ xx =(h h − h h )+(h h + h h )e, 1 2 2 1 1 2 1 2 ̄ ̄ where product of quaternions is used on the right hand side and h , h are the conjugates 1 2 � � of h , h ∈ ℍ . Like for quaternions, the conjugation x ̄ = h − h e in  relates with the non- 1 2 1 2 � � �� � �� � �� commutativity: xx = x ̄ x ̄ . One has also the associator [x, x , x ]=(xx )x − x(x x ) , that � �� vanishes whenever two among x, x , x ∈  are equal or conjugate. The identification x = h + h e ∈ 𝕆 ↔ (h , h )∈ ℍ , used in the previous formula, is not 1 2 1 2 an isomorphism of (left or right) quaternionic vector spaces. To get an isomorphism one has instead to go through the following hypercomplex structure (I,  J,  K) on ℝ ≅ 𝕆 . For x = h + h e ∈  , where (h , h )∈ ℍ , define 1 2 1 2 I(x)= x ⋅ i, J(x)= x ⋅ j, K(x) =(x ⋅ i) ⋅ j or equivalently I(h , h )=(h i, −h i), J(h , h ))=(h j, −h j), K(x) =(h k, h k). 1 2 1 2 1 2 1 2 1 2 This observation goes likely back to the very discovery of octonions in the mid-1800s. The alternative approach to the same isomorphism used in our Introduction does not seem however to have appeared before 1989, when R. Bryant and R. Harvey [6] looked at the map L ∶ ℍ → 𝕆, L(h , h )= h +(kh k)e ∈ 𝕆 , 1 2 1 2 and observed it satisfies ̄ ̄ L[(h , h )i]= h i +(kh ik)e, L[(h , h )j]= h j +(kh jk)e, L[(h , h )k]= h k +(kh )e. 1 2 1 2 1 2 1 2 1 2 1 2 x = h , x = kh k x = x + x e This, in terms of and of the octonion , can be read exactly 1 1 2 2 1 2 as in our previous approach: L[(h , h )i]= x ⋅ i, L[(h , h )j]= x ⋅ j, L[(h , h )k] =(x ⋅ i) ⋅ j, 1 2 1 2 1 2 and as mentioned L Φ =Φ . Spin(7) K 3 The quaternionic 4‑form and the Cayley calibration in  ℝ A possible way to produce 4-forms canonically associated with some G-structures is through the notion of Clifford system. We recall the definition, originally given in the con- text of isoparametric hypersurfaces, cf. [9]. Definition 3.1 A Clifford system on a Riemannian manifold (M,  g) is a vector sub-bundle E ⊂ End TM locally spanned by self-adjoint anti-commuting involutions I ,… , I . Thus 1 r 2 ∗ I = Id, I = I , I ◦I =−I ◦I , and the I are required to be related, in the inter- sections of trivializing sets, by matrices of SO(r) . The rank r of E is said to be the rank of the Clifford system. Possible ranks of irreducible Clifford systems on ℝ are classified, up to N = 32 , in Table 1. 1 3 Clifford systems, Clifford structures, and their canonical… 105 In particular, the Clifford system of rank 3 in ℝ can be defined by the classical Pauli matrices: 01 0 − i 10 I = , I = , I = ∈ U(2) ⊂ SO(4), 1 2 3 10 i 0 0 − 1 and the Clifford system of rank 5 in ℝ by the following similar (right) quaternionic Pauli matrices: 0 Id 0 − R 0 − R i j I = , I = , I = , 1 2 3 Id 0 R 0 R 0 i j (3.1) 0 − R Id 0 I = , I = ∈ Sp(2) ⊂ SO(8), 4 5 R 0 0 − Id 2 8 where as before R , R , R denote the multiplication on the right by i, j, k on ℍ ≅ ℝ . i j k According to Table 1, there is also a Clifford system with r = 4 in ℝ , explicitly defined by selecting e.g. 0 Id 0 − R 0 − R 0 − R i j k I = , I = , I = , I = . 1 2 3 4 Id 0 R 0 R 0 R 0 i j k Going back to rank r = 5 , from the quaternionic Pauli matrices I , I , I , I , I , one gets 1 2 3 4 5 the 10 complex structures on ℝ I = I ◦I for 1 ≤ 𝛼 <𝛽 ≤ 5. 𝛼𝛽 𝛼 𝛽 Their Kähler forms give rise to a 5 × 5 skew-symmetric matrix =( ), and one can easily see that both the following matrices of Kähler 2-forms ⎛ ⎞ L L i j ⎜ ⎟ =( )∈ (5) and  = − 0  ∈ (3) R  L L L i k ⎜ ⎟ − −  0 ⎝ ⎠ L L j k allow to write the (left) quaternionic 4-form of ℍ as 2 2 2 2 Ω =− 𝜃 =[𝜔 + 𝜔 + 𝜔 ]. 𝛼𝛽 L L L (3.2) i j k 𝛼<𝛽 On the other hand, as mentioned in the Introduction, the subgroup Spin(7) ⊂ SO(8) (generated by the right translation R , u ∈ S ⊂ Im ) gives rise to the Cayley calibration Φ ∈Λ : Spin(7) Table 1 Rank of irreducible Dimension N 2 4 8 8 16 16 16 16 32 64 64 ... Clifford systems in ℝ Rank r 2 3 4 5 6 7 8 9 10 11 12 ... 1 3 106 K. B. M. Boydon, P. Piccinni 1 1 2 2 2 2 Φ =− [𝜙 + 𝜙 +… 𝜙 ]= 𝜁 . Spin(7) i j h 𝛼𝛽 6 6 𝛼<𝛽 Here  ,  ,  ,  ,  ,  ,  . are the Kähler 2-forms associated with the complex struc- i j k e f g h tures (J , J , J , J , J , J , J )= (R , R , R , R , R , R , R ) , and  =( )∈ (7) is the 1 2 3 4 5 6 7 i j k e f g h matrix of the Kähler 2-forms of compositions J = J ◦J . It is worth to recall that under the action of Sp(2)Sp(1), the space of exterior 2-forms 2 8 Λ ℝ decomposes as 2 2 2 2 Λ =Λ ⊕ Λ ⊕ Λ , 10 15 3 (cf. [22, page 125] as well as [23, page 93]). Here lower indices denote the dimensions of irreducible components. Here Λ ≅ (2) is generated by the Kähler forms  of the J (𝛼 <𝛽 ) , compositions of the five quaternionic Pauli matrices, and Λ ≅ (1) is gener- 𝛼𝛽 ated by the Kähler forms  , , . L L L i j k By denoting by  the second coefficient in the characteristic polynomial of the involved skew-symmetric matrices, we can rewrite formula (3.2) of Ω as: Ω =−  ( )=  ( ), L 2 R 2 L ⎛ L L ⎞ i j ⎜ ⎟ = − 0  ∈ (3) where  =( )∈ (5) , and . L L R  L i k ⎜ ⎟ ⎝− −  0 ⎠ L L j k Similarly, under the Spin(7) action one gets the decomposition: 2 2 2 Λ =Λ ⊕ Λ , 7 21 see [13, page 256] or [14, page 47]. Here Λ is generated by the Kähler forms  of J = R , R ,… , R and Λ ≅ (7) is generated by the Kähler forms  of the the i j h J ◦J (𝛼 <𝛽 ) . Thus, in the  notation: 𝛼 𝛽 2 1 1 Φ =−  =  (),  =( )∈ (7). Spin(7) 2 6 6 All the exterior 4-forms ,  and the have been studied systematically as calibra- tions in the space ℝ , cf. [8]. 4 Proof of Theorems 1.1 and 1.2 The matrix  =( )∈ (5) in the statement of Theorem  1.2 is defined as follows. Let I ( = 1, … ,5) be the left quaternionic Pauli matrices defined as in (3.1) but by using the L L L I = I ◦I left quaternionic multiplications L , L , L by i, j.k. If and if  are the Kähler i j k 2-forms associated to I , a computation shows that 2 2 2 2 2 2 2 2 2 2 2Ω =  +  +  +  +  +  +  +  +  +  , 12 13 24 34 23 14 15 25 35 45 and note the symmetry with the first identity in formula (3.2). We express now the 2-forms  and  in the coordinates of ℝ , using the following 1 8 8 {dx , … , dx } ⊂ Λ ℝ be the standard basis of 1-forms in ℝ . Then abridged notations. Let 1 8 1 3 Clifford systems, Clifford structures, and their canonical… 107 (scriptsize) denotes dx ∧ dx and  denotes dx ∧ dx ∧ dx ∧ dx , and ⋆ denotes the Hodge star, so that a + ⋆ = a + ⋆a . One gets: =−12 + 34 + 56 − 78, =−13 − 24 + 57 + 68, =−14 + 23 + 58 − 67, (4.1) =−14 + 23 − 58 + 67, =+13 + 24 + 57 + 68, =−12 + 34 − 56 + 78, and =−15 − 26 − 37 − 48, =−16 + 25 + 38 − 47, (4.2) =−17 − 28 + 35 + 46, =−18 + 27 − 36 + 45, so that, if  =( ), 2 2 2 𝜏 (𝜃 )= 𝜃 + 𝜃 +⋯ + 𝜃 12 13 45 (4.3) =−121234 − 41256 − 41357 + 41368 − 41278 − 41467 − 41458 + ⋆ =−2Ω . Next: =−12 − 34 + 56 + 78, =−13 + 24 + 57 − 68, =−14 − 23 + 58 + 67, =+14 + 23 + 58 + 67, =−13 + 24 − 57 + 68, (4.4) =+12 + 34 + 56 + 78, =−15 − 26 − 37 − 48, =−16 + 25 − 38 + 47, =−17 + 28 + 35 − 46, =−18 − 27 + 36 + 45, and, if  =( ), 2 2 2 𝜏 (𝜂 )= 𝜂 + 𝜂 +⋯ + 𝜂 12 13 45 (4.5) = 121234 − 41256 − 41357 − 41368 + 41278 + 41467 − 41458 + ⋆ =−2Ω . Similarly: 1 3 108 K. B. M. Boydon, P. Piccinni =−12 + 34 + 56 − 78, =−13 − 24 + 57 + 68, =−14 + 23 + 58 − 67, =−15 − 26 − 37 − 48, e (4.6) =−16 + 25 − 38 + 47, =−17 + 28 + 35 − 46, =−18 − 27 + 36 + 45, and one easily deduces also formulas for the  (cf. [3, 19]). Then, by (1.2): Φ = 1234 + 1256 + 1357 + 1368 − 1278 − 1467 + 1458 + ⋆. Spin(7) (4.7) By computing the squares of the 2-forms in (4.6) (4.4) and comparing with Formulas (4.5), (4.3), (4.7), the identities listed in Theorems 1.1 and 1.2 are recognized. 5 Even Clifford structures in dimension 8 We recall first the following notion, proposed in 2011  by A. Moroianu and U. Semmel- mann, [17]. Definition 5.1 Let (M,  g) be a Riemannian manifold. An even Clifford structure is the choice of an oriented Euclidean vector bundle E of rank r ≥ 2 over M, together with a bundle morphism  from the even Clifford algebra bundle even r 2 r − ∶ Cl E → EndTM such that Λ E ↪ End TM. r is called the rank  of the even Clifford structure. The even Clifford structure E is said to be parallel if there exists a metric connection ∇ on E such that  is connection preserving, i.e. (∇ )= ∇ (), X X even g for every tangent vector X ∈ TM and section  of Cl E , where ∇ is the Levi Civita connection. Rank 2, 3, 4 parallel even Clifford structures are equivalent to complex Kähler, quater - nion Kähler, product of two quaternion Kähler. Besides them, higher rank parallel non-flat even Clifford structures in dimension 8 are listed in Table 2, cf. [17]. A class of examples of even Clifford structures are those coming from Clifford systems as defined in Sect.  3. Namely, if the vector sub-bundle E ⊂ End TM , locally spanned by self-adjoint anticommuting involutions I ,… , I , defines the Clifford system, then 1 r one easily recognizes that through the compositions I = I ◦I , the Clifford morphism even ∶ Cl (E ) → End TM is well defined. An example is given by the first row of Table  2, where the quaternion Kähler structure is constructed via the local I = I ◦I defined as in Sect.  3, by using on the model space ℍ the quaternionic Pauli matrices. The remaining three rows of Table  2  correspond to 1 3 Clifford systems, Clifford structures, and their canonical… 109 Table 2 Parallel non-flat even r M Clifford structures of rank ≥ 5 in M 5 Quaternion Kähler 6 Kähler Spin(7) 7 holonomy 8 Riemannian Table 3 Generators and associated 4-forms r M Clifford bundle generators Associated 4-form 1 1 5 qK I , I , I , I , I 1 2 3 4 5 Ω =−  ( ), Ω =−  ( ) L 2  R 2 2 2 6 Kähler J , J , J , J , J , J Φ =  ( )=−5 1 2 3 4 5 6 Spin(6) 2 1 1 7 Spin(7) hol J , J , J , J , J , J , J 1 2 3 4 5 6 7 Φ =−  =  ( ) Spin(7) 2 6 6 8 Riemannian I, J , J , J , J , J , J , J Φ =  ( )= 0 1 2 3 4 5 6 7 SO(8) 2 essential even Clifford structures, i. e. to even Clifford structures that cannot be defined throw a Clifford system, cf. [21] for a discussion on this notion. Table  3 gives a description of the Clifford bundle generators and of the canonically associated 4-form for each of the four even Clifford structures on ℝ . Here I , I , I , I , I are the (left or right) quaternionic Pauli matrices, ( ) , ( ) are 1 2 3 4 5 like in Sect.  4. Notations (J , J , J , J , J , J , J )= (R , R , R , R , R , R , R ) are also used, 1 2 3 4 5 6 7 i j k e f g h is the Kähler form of J and is the Kähler form of J ◦J . Finally , ( )∈ (8) , with entries ± in the first line and column and with entries  ∈ (7). It is of course desirable to give examples of Riemannian manifolds (M , g) supporting both a Sp(2) ⋅ Sp(1) and a Spin(7) structure. Rarely the metric g can be the same for both structures, but this is possible of course for parallelizable (M , g) . On this respect, homoge- neous (M , g) with an invariant Spin(7) structure have been recently classified [1], by mak - ing use of the following topological condition for compact oriented spin M : p (M)− 4p (M)+ 8(M)= 0. 7 1 Some of the obtained examples are parallelizable, e. g. diffeomorphic to S × S and 5 3 5 3 S × S . On the latter, S × S , using two natural parallelizations, one can define two Spin(7) structures, both of general type (in the 1986 M. Fernández Spin(7) framework), and the hyperhermitian structure associated with one of them corresponds to a family of Calabi- Eckmann [20]. To get examples of 8-dimensional manifolds that admit both a locally conformally hyperkähler metric g and a locally conformal parallel Spin(7) metric, that is either the same g as before, or a different metric g , a good point to start with is the class of compact 7 7 3-Sasakian 7-dimensional manifolds (S , g) . Many examples of such (S , g) and with arbi- trary second Betti numbers have been given by Ch. Boyer, K. Galicki et al, cf. [5]. In par- ticular, recall that given the 3-Sasakian (S , g) one gets a locally conformally hyperkähler 7 1 metric g on the product S × S [18]. This can also be expressed by saying that the 3-Sasa- kian metric g has the property of being nearly parallel G , and in particular with 3 linearly independent Killing spinors, cf. [4, pp. 536–538]. Moreover the differentiable manifold S 1 3 110 K. B. M. Boydon, P. Piccinni admits, besides the 3-Sasakian metric g, another metric g that is also nearly parallel G but proper, i. e. with only one non zero Killing spinor. This allows to extend the metrics g and g to the product with S and to get both the properties of locally conformally hyperkähler 8 7 1 and locally conformally parallel Spin(7) on (M , g)=(S × S , g) and of locally confor- 8 � 1 � mally parallel Spin(7) on (M , g )=(S × S , g ) , cf. also [12]. Further examples of 8-dimensional differentiable manifolds admitting both a Sp(2) ⋅ Sp(1)-structure with respect to a metric g and a Spin(7)-structure with respect to a metric g include the Wolf spaces ℍP and G ∕SO(4) , cf. [1]. Finally, the non singular sex- 5 6 6 tic Y = {[z , … , z ]∈ ℂP , z +⋯ + z = 0} is also an example, where a metric g giving 0 5 0 5 an almost quaternionic structure is insured by a result in [7], and a metric g with holonomy SU(4) ⊂ Spin(7) by Calabi-Yau theorem, cf. [13, p. 139]. 6 Dimension 16 2 16 A Clifford system with r = 9 in 𝕆 ≅ ℝ is given by the following octonionic Pauli matrices: 0 Id 0 − R 0 − R i j I = , I = , I = , 1 2 3 Id 0 R 0 R 0 i j 0 − R 0 − R 0 − R k e f I = , I = , I = , 4 5 6 R 0 R 0 R 0 k e f 0 − R 0 − R Id 0 g h I = , I = , I = ∈ SO(16), 7 8 9 R 0 R 0 0 − Id g h and of course now R , R ,… , R denote the multiplication on the right by the unit octonions i j h 2 16 i, j,… , h on 𝕆 ≅ ℝ . Looking back at Table 1, we see that in ℝ there are also irreducible Clifford systems with r = 8, 7, 6 . According to [21], convenient choices are the following: r = 8 ∶ I ,… , I , r = 7 ∶ I ,… , I , r = 6 ∶ I , I , I , I , I , I . 1 8 2 8 1 2 3 4 5 9 It is now worth to remind the following parallel situations in complex, quaternionic and octonionic geometry. The groups , Sp(2) ⋅ Sp(1) ⊂ SO(8) , Spin(9) ⊂ SO(16) U(2) ⊂ SO(4) are the stabilizers of the vector subspaces 3 + 4 5 + 8 9 + 16 E ⊂ End (ℝ ), E ⊂ End (ℝ ), E ⊂ End (ℝ ) spanned respectively by the Pauli, quaternionic Pauli, octonionic Pauli matrices. Moreover, U(2) , Sp(2) ⋅ Sp(1) , Spin(9) are symmetry groups of the Hopf fibrations respectively: 1 3 7 S S S 3 2 1 7 4 1 15 8 1 S ⟶ S ≅ ℂP , S ⟶ S ≅ ℍP , S ⟶ S ≅ 𝕆 P . 2 2 4 2 8 2 Finally, U(2) , Sp(2) ⋅ Sp(1) , Spin(9) are stabilizers in Λ ℂ ,Λ ℍ ,Λ 𝕆 of the following canonically associated forms, cf. [2]: ∗ 2 ∗ 4 ∗ 8 Φ = p  d𝓁 ∈Λ , Φ = p  d𝓁 ∈Λ , Φ = p  d𝓁 ∈Λ , U(2) 𝓁 Sp(2)⋅Sp(1) 𝓁 Spin(9) 𝓁 𝓁 𝓁 𝓁 1 1 1 ℂP ℍP 𝕆 P 1 3 Clifford systems, Clifford structures, and their canonical… 111 def def 2 2 2 where  is the volume form on the line or in ℂ or ℍ or  , = {(0, y)} = {(x, mx)} 2 4 2 8 2 16 p ∶ ℂ ≅ ℝ or ℍ ≅ ℝ or 𝕆 ≅ ℝ ⟶ is the projection on the line ) , and note that the integral formula is based on the volume of distinguished planes. In the three cases one gets in this way the Kähler 2-form of ℂ , the 2 2 quaternion Kähler 4-form of ℍ and the canonical 8-form of  . 7 Rank 8, 7 and 6 Clifford systems on  ℝ Look now closer at the nine octonionic Pauli matrices, that define a rank 9 Clifford system in ℝ , and at the choices among them that give rise to ranks r = 8, 7, 6 (cf. previous Sec- tion). The compositions I = I ◦I , 𝛼 <𝛽 , for all choices r = 6, 7, 8, 9 are bases of the 𝛼𝛽 𝛼 𝛽 Lie algebras (6) ⊂ (7) ⊂ (8) ⊂ (9) ⊂ (16). Like in the previous Sections, we can write the matrices of Kähler forms  associated to I , and we use for them the following notations: A B C D =( )∈ (6),  =( )∈ (7),  =( )∈ (8),  =( )∈ (9). The second coefficients  of their characteristic polynomial give rise to the following invariant 4-forms A B C D 4 16 ( ),  ( ),  ( ),  ( )∈ Λ ℝ 2 2 2 2 that can be written in (the differentials of) the coordinates of ℝ = 𝕆 ⊕ 𝕆 : 1,2,3,4,5,6,7,8;1 ,2 ,3 ,4 ,5 ,6 ,7 ,8 . We recall in particular that in the Spin(9) situation, the following identity holds: ( )= 0, D 8 and this gives evidence to the next coefficient  ( )∈Λ , proportional to the 8-form Φ , as studied in [19]. Spin(9) 8 The 4‑forms 8 and 8 Spin(8) Spin(7)U(1) Look now only at I ,… , I and at the matrix 1 8 =( )∈ (8) of Kähler forms associated to I = I ◦I . By using coordinate 1-forms 1,2,3,4,5,6,7,8;1 ,2 ,3 ,4 ,5 ,6 ,7 ,8 , an explicit computation, based on the formulas in [19] for the  , yields: 1 3 112 K. B. M. Boydon, P. Piccinni 1 1 2 Φ = 𝜏 (𝜓 )= 𝜓 = 1234 + 1256 − 1278 + 1357 + 1368 + 1458 Spin(8) 2 𝛼𝛽 4 4 1=𝛼<𝛽 4 16 � � − 1467 − 2358 + 2367 + 2457 + 2468 − 3456 + 3478 + 5678 + aba b ∈Λ ℝ . 1=a<b Here boldface notations have the following meaning: � � � � � � � � � � � � abcd = abc d − ab cd + ab c d + a bcd − a bc d + a b cd. By excluding now the Kähler forms involving I and I , one gets the matrix 1 9 =( )∈ (7) . Now similar computations lead to: Φ = 𝜏 (𝜓 )= 6[1234 + 1256 Spin(7)U(1) 2 − 1278 + 1357 + 1368 + 1458 − 1467 − 2358 + 2367 + 2457 + 2468 − 3456 + 3478 + 5678] � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � + 6[1 2 3 4 + 1 2 5 6 − 1 2 7 8 + 1 3 5 7 + 1 3 6 8 + 1 4 5 8 − 1 4 6 7 − 2 3 5 8 � � � � � � � � � � � � � � � � � � � � � � � � � + 2 3 6 7 + 2 4 5 7 + 2 4 6 8 − 3 4 5 6 + 3 4 7 8 + 5 6 7 8]+ 6 aba b 1=a<b + 2[1234 + 1256 − 1278 + 1357 + 1368 + 1458 − 1467 − 2358 4 16 + 2367 + 2457 + 2468 − 3456 + 3478 + 5678] ∈Λ ℝ , where boldface notations have the same meaning as before. The presence of the factor U(1) in the group Spin(7)U(1) is here due to a computation showing that matrices in SO(16) commuting with the seven involutions I ,… I are a U(1) subgroup, well identified in [3, 2 8 Chapter 6, p. 44] (cf. also proof of Theorem 1.4 below). We take this opportunity to remark that the expression of Φ written in the paper Spin(7)U(1) [21] contains some errors in the coefficients as well as some missing terms, and has to be corrected by the present one. Note also that Φ restricts, on any of the two sum- Spin(7)U(1) 16 8 8 mands of ℝ = ℝ ⊕ ℝ , and up to a factor 6, to the usual Cayley calibration of [11]. We are now ready for the proofs of Theorems 1.3 and 1.4. The following notion has already implicitly introduced in the statement of Theorem 1.3. 16 2 � Definition 8.1 Let P be a 4-plane in the real vector space ℝ ≅ 𝕆 = 𝕆 ⊕ 𝕆 , and let 2 � 2 � ∶  →  and  ∶  →  be the orthogonal projections to  and  . P is said to be a transversal Cayley 4-plane if both its projections (P) ,  (P) are 2-dimensional and invari- ant under a same complex structure u ∈ S ⊂ Im. Proof of Theorem 1.3 Recall that Spin(8) can be characterized as the subgroup of the fol- lowing matrices A ∈ SO(16): a 0 A = 0 a where a , a ∈ SO(8) are triality companions, i. e. and for any v ∈  there exists a w ∈ + − such that R = a R a (cf. [10, p. 278-279]). It follows that Spin(8) contains the diagonal w + v Spin(7) (characterized by choices a = a ) and acts transitively on transversal 4-planes Δ + − of ℝ . On the other hand the 4-form Φ is invariant under the action of Spin(8) . Thus, Spin(8) since Φ takes value 1 on the 4-plane spanned by the coordinates 121 2 , Φ takes Spin(8) Spin(8) value 1 on any tranversal Cayley 4-plane in ℝ . ◻ 1 3 Clifford systems, Clifford structures, and their canonical… 113 Next, let Q be any 4-plane of ℝ . By looking at the expression of Φ , we see that Spin(8) the only possibilities for having non zero value on Q are that (Q) and  (Q) are 2-dimen- sional. For such 4-planes Q we can use the following canonical form with respect to the complex structure i ∈ S : � � � � � Q = e ∧(R e cos 𝜃 + e sin 𝜃 ) ⊕ e ∧(R e cos 𝜃 + e sin 𝜃 ) , 1 i 1 2 i 1 1 2 where the pairs e , e and e , e are both orthonormal and respectively in  and in  , and 1 2 1 2 0 ≤  ≤ with angles limited by and  ≤  ≤  −  . The above canonical form for Q is a small variation of the canonical forms that are used in a proof of the classical Wirtinger’s inequaliy (cf. [16, p.6]) and in characterizations of Cayley 4-planes in ℝ in the Harvey- Lawson foundational paper (cf. [11, p. 121]). Its proof follows the steps of proof of the mentioned canonical form, as explained in details in [16]. From this canonical form we see that Φ (Q) ≤ 1 for any 4-plane Q, and that the equality holds only if  =  = 0 , i. e. Spin(8) for transversal Cayley 4-planes. Proof of Theorem 1.4 The leading terms in the expression of Φ are those with coef- Spin(7)U(1) ficient 6, thus terms involving only coordinates among 12345678 , or only coordinates among 1 2 3 4 5 6 7 8 , or terms aba b . Look first at the first and second types of terms. We already 2 � =  ⊕ mentioned that the restriction of Φ to any of the summands in is Spin(7)U(1) the usual Cayley calibration in ℝ , whose calibrated 4-planes are the Cayley planes. Thus, for the first two types of terms, we get as calibrated 4-planes just the Cayley 4-planes that are contained in the octonionic lines with slope m = 0 and m =∞ . In the remaining case of terms aba b one gets as calibrated 4-planes the transversal Cayley 4-planes that are con- tained in the octonionic line  (leading coefficient m = 1 ). Now Spin(7) acts on the indi- ,  , vidual octonionic lines , and the only possibility to move planes out of them is 0 1 ∞ through the factor U(1) . In fact, the discussion in [3, Chapter 6, p. 44] shows that the factor U(1) in the group Spin(7)U(1) moves the octonionic lines through the circle, contained in the space S of the octonionic lines, passing through the three points m = 0, 1,∞ . This cor- responds to admitting any real coefficient: m ∈ ℝ ∪∞ as slope of the octonionic lines that are admitted to contain the calibrated 4-planes. ◻ Remark 8.2 Following the recent work [15] by J. Kotrbatý, one can use octonionic 1-forms, according to the following formal definitions: dx = d + id + jd + kd + ed + fd + gd + hd, dx = d − id − jd − kd − ed − fd − gd − hd, � � � � � � � � � dx = d + id + jd + kd + ed + fd + gd + hd , � � � � � � � � dx = d − id − jd − kd − ed − fd − gd − hd , � 16 referring to pairs of octonions (x, x )∈ 𝕆 ⊕ 𝕆 = ℝ . Then, in the same spirit proposed in [15], a straightforward computation yields the following formula, much simpler way to write the Spin(8) canonical 4-form of ℝ : Φ = (dx ∧ dx )∧(dx ∧ dx). Spin(8) Similarly, one gets that the Spin(7)U(1) canonical 4-form of ℝ can be written in octon- ionic 1-forms as: 1 3 114 K. B. M. Boydon, P. Piccinni 2 � 2 Φ = (dx ∧ dx) +(dx ∧ dx ) Spin(7)U(1) � 2 2 � � � − (dx ∧ dx ) +(dx ∧ dx) − (dx ∧ dx )∧(dx ∧ dx) . Details of both computations are in [3]. Acknowledgements Kai Brynne M. Boydon was supported by University of the Philippines OVPAA Doc- toral Fellowship. Part of the present work was done during her visit at Sapienza Università di Roma in the academic year 2018–19, and she thanks Sapienza University and Department of Mathematics “Guido Castelnuovo” for hospitality. Paolo Piccinni was supported by the group GNSAGA of INdAM, by the PRIN Project of MIUR “Real and Complex Manifolds: Topology, Geometry and holomorphic dynamics”, and by Sapienza Università di Roma Project “Polynomial identities and combinatorial methods in algebraic and geometric structures”. Funding Open access funding provided by Università degli Studi di Roma La Sapienza within the CRUI- CARE Agreement. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Com- mons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creat iveco mmons .org/licen ses/by/4.0/. References 1. Alekseevsky, D.V., Chrysikos, I., Fino, A., Raffero, A.: Homogeneous 8-manifolds admitting invariant Spin(7)-structures. Int. J. Math. 31(8), 2050060 (2020)  2. Berger, M.: Du côté de chez Pu. Ann. Sci. École Norm. Sup. 4(5), 1–44 (1972) 3. Boydon, K.B.: Clifford Systems and Clifford Structures with their canonical associated 4-forms in dimensions 8 and 16. Dissertation for the Degree of Doctor of Philosophy in Mathematics, University of the Philippines Diliman (2020) 4. Boyer, Ch. P., Galicki, K.: 3-Sasakian manifolds. Surveys in differential geometry vol. VI: essays on Einstein manifolds, pp. 123–184. Int. Press, Boston, MA (1999) 5. Boyer, C.P., Galicki, K., Mann, B.M., Rees, E.: Compact 3-Sasakian 7-manifolds with arbitrary second Betti number. Invent. Math. 131, 321–344 (1998) 6. Bryant, R.L., Harvey, R.: Submanifolds in hyper-Kähler Geometry. J. Am. Math. Soc. 2(1), 1–31 (1989) 7. Čadek, K., Vanžura, J.: Almost quaternionic structures on eight-manifolds. Osaka J. Math. 35(1), 165– 190 (1998) 8. Dadok, J., Harvey, R., Morgan, F.: Calibrations on ℝ . Trans. Am. Math. Soc. 307, 1–40 (1988) 9. Ferus, D., Karcher, H., Münzner, H.F.: Cliffordalgebren und neue isoparametrische Hyperflächen. Math. Z. 177, 479–502 (1981) 10. Harvey, F.R.: Spinors and Calibrations. Academic Press Inc., Cambridge (1990) 11. Harvey, R., Lawson Jr., H.B.: Calibrated geometries. Acta Math. 148, 47–157 (1982) 12. Ivanov, S., Parton, M., Piccinni, P.: Locally conformal parallel and Spin(7) manifolds. Math. Res. Lett. 13(2–3), 167–177 (2006) 13. Joyce, D.D.: Compact Manifolds with Special Holonomy. Oxford University Press, Oxford (2000) 14. Joyce, D.D.: Compact Riemannian manifolds with exceptional holonomy. In: LeBrun, C., Wang, M. (eds.) Surveys in Differential Geometry: Essays on Einstein Manifolds, pp. 39–65. International Press, Vienna (1999) 15. Kotrbatý, J.: Octonion-valued forms and the canonical 8-form on Riemannian manifolds with a Spin(9)-structure. J. Geom. Anal. 30(4), 3616–3640 (2020) 1 3 Clifford systems, Clifford structures, and their canonical… 115 16. Lotay, J.D.: Calibrated Submanifolds. arXiv :1810.08709 v1 (2018) 17. Moroianu, A., Semmelmann, U.: Clifford structures on Riemannian manifolds. Adv. Math. 228, 940– 967 (2011) 18. Ornea, L., Piccinni, P.: Locally conformal Kähler structures in quaternionic geometry. Trans. Am. Math. Soc. 349(2), 641–655 (1997) 19. Parton, M., Piccinni, P.: Spin(9) and almost complex structures on 16-dimensional manifolds. Ann. Glob. Anal. Geom. 41(3), 321–345 (2012) 20. Parton, M., Piccinni, P.: Parallelizations on products of spheres and octonionic geometry. Complex Manifolds Spec. Issue Complex Geom. Lie Groups 6, 138–149 (2019) 21. Parton, M., Piccinni, P., Vuletescu, V.: Clifford systems in octonionic geometry. Rend. Sem. Mat. Univ. Pol. Torino Workshop Sergio Console 74, 267–288 (2016) 22. Salamon, S.M.: Riemannian Geometry and Holonomy Groups. Longman Sc and Tech., Harlow (1989) 23. Salamon, S.M.: Quaternionic-Kähler geometry. In: LeBrun, C., Wang, M. (eds.) Surveys in Differen- tial Geometry: Essays on Einstein Manifolds, pp. 83–121. International Press, Vienna (1999) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. 1 3 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Springer Journals

Clifford systems, Clifford structures, and their canonical differential forms

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2 8 A comparison among different constructions in ℍ ≅ ℝ of the quaternionic 4-form Φ and of the Cayley calibration Φ shows that one can start for them from the Sp(2)Sp(1) Spin(7) same collections of “Kähler 2-forms”, entering both in quaternion Kähler and in Spin(7) geometry. This comparison relates with the notions of even Clifford structure and of Clif- ford system. Going to dimension 16, similar constructions allow to write explicit formulas in ℝ for the canonical 4-forms Φ and Φ , associated with Clifford systems Spin(8) Spin(7)U(1) related with the subgroups Spin(8) and Spin(7)U(1) of . We characterize the cali- SO(16) brated 4-planes of the 4-forms Φ and Φ , extending in two different ways the Spin(8) Spin(7)U(1) notion of Cayley 4-plane to dimension 16. Keywords Octonions · Clifford system · Clifford structure · Calibration · Canonical form Mathematics Subject Classification Primary 53C26 · 53C27 · 53C38 1 Introduction In 1989 R. Bryant and R. Harvey defined the following calibration, of interest in hyper - kähler geometry [6]: 1 1 1 2 2 2 4 n Φ =−  −  +  ∈Λ ℍ . R R R i j k 2 2 2 Communicated by Vicente Cortés. * Paolo Piccinni piccinni@mat.uniroma1.it Kai Brynne M. Boydon kbboydon@math.upd.edu.ph Institute of Mathematics, University of the Philippines Diliman, Quezon City, Philippines Dipartimento di Matematica, Sapienza Università di Roma, Piazzale Aldo Moro 2, 00185 Roma, Italy 1 3 Vol.:(0123456789) 102 K. B. M. Boydon, P. Piccinni In this definition, ( , , ) are the Kähler 2-forms of the hypercomplex structure R R R i j k (R , R , R ) , defined by multiplications on the right by unit quaternions (i,  j, k) on the space i j k 4n n ℝ ≅ ℍ . When n = 2 , the Bryant-Harvey calibration Φ relates with Spin(7) geometry. This is easily recognized by using the map L ∶ ℍ → 𝕆 , L(h , h )= h +(kh k)e ∈ 𝕆 , 1 2 1 2 from pairs of quaternions to octonions, that yields the identity L Φ =Φ . Spin(7) K (1.1) 4 8 Here Φ ∈Λ ℝ is the Spin(7) 4-form, or Cayley calibration, studied since the R. Har- Spin(7) vey and H. B. Lawson’s foundational paper [11], and defined through the scalar product and the double cross product of ℝ ≅ 𝕆 : Φ (x, y, z, w)= < x , y × z × w > = < x , y(zw ̄ ) >, Spin(7) assuming here orthogonal y, z, w ∈ . The present paper collects some of the results in the first author Ph.D. thesis [3], inspired from viewing formula (1.1) as a way of constructing the Cayley calibration Φ Spin(7) through the 2-forms  , , . As well known, by summing the squares of the latter R R R i j k 2-forms one gets another remarkable calibration, namely the quaternionic right 4-form Ω . Thus  , , , somehow building blocks for quaternionic geometry, enter also in R R R i j k Spin(7) geometry. A first result is the following Theorem  1.1, a kind of “other way around” of formula (1.1). To state it, recall that the Cayley calibration Φ can also be constructed as sum of Spin(7) squares of “Kähler 2-forms” associated with complex structures on ℝ ≅ 𝕆 , defined by the unit octonions. In fact, cf. [19, Prop.10]: 1 1 2 2 2 2 2 2 2 Φ =− ( +  +⋯ +  )= ( +  +  +⋯ +  ). (1.2) Spin(7) i j h ij ik ik gh 6 6 ,  ,… ,  are the Kähler 2-forms associated with the 7 complex structures Here i j h R , R ,… , R on ℝ ≅ 𝕆 , the right multiplications by the unit octonions i,  j,  k,  e,  f,  g,  h, i j h ,  ,… ,  are the Kähler 2-forms associated with the 21 complex structures and ij ik gh R = R ◦R , R = R ◦R ,… , R = R ◦R , compositions of them. ij i j ik i k gh g h 4 2 Theorem 1.1 The right quaternionic 4-form Ω ∈Λ ℍ can be obtained from the Kähler forms  ,  ,… ,  associated with the complex structures R , R ,… , R as: i j h i j h 2 2 2 2 2 2 2 Ω = 2[ +  +  −  −  −  −  ]. i j k e f g h Moreover, by selecting any five out of the seven (J = R , J = R , … , J = R ) and by look- 1 i 2 j 7 h ing at the matrix  =( )∈ (5) of Kähler 2-forms of their compositions J = J ◦J , one can get the left quaternionic 4-form Ω as Ω =− 𝜁 , 𝛼𝛽 𝛼<𝛽 up to a permutation or change of signs of some coordinates in ℝ . 1 3 Clifford systems, Clifford structures, and their canonical… 103 On the same direction as in Bryant-Harvey’s formula (1.1), a similar result is the follow- ing (cf. Sect. 4 for more details): 4 8 Theorem  1.2 The Cayley calibration Φ ∈Λ ℝ can be obtained from the Kähler Spin(7) L L L L L 2-forms  (1 ≤ 𝛼 <𝛽 ≤ 5) associated to complex structures I = I ◦I , where I ,… I 1 5 are anti-commuting self-dual involutions in ℝ . Namely: 2 2 2 2 2 2 2 2 2 2 Φ =  +  +  +  −  −  −  −  −  −  , Spin(7) 12 13 24 34 23 14 15 25 35 45 and on the other hand one can get the right quaternion Kähler as: 2 2 2 2 2 2 2 2 2 2 Ω =−  +  +  +  +  +  +  +  +  +  . 12 13 24 34 23 14 15 25 35 45 Moving to dimension 16 and in Sect. 6, we consider two exterior 4-forms 4 16 Φ , Φ ∈Λ ℝ , Spin(8) Spin(7)U(1) canonically associated with subgroups Spin(8), Spin(7)U(1) ⊂ SO(16) , and we write their explicit expressions in the 16 coordinates. We will see in the next two statements which 4-planes of ℝ are calibrated by Φ and by Φ . Spin(8) Spin(7)U(1) The 4-forms Φ , Φ and the respective calibrated 4-planes can be compared Spin(8) Spin(7)U(1) with other calibrations in ℝ , in particular with the previously mentioned Bryant-Harvey 4-form Φ . It is thus appropriate to remind the main theorem in [6, Theorem 2.27], namely n 4n that, in any ℍ ≅ ℝ , the Bryant-Harvey 4-form Φ calibrates the Cayley 4-planes that are 2 n contained in a quaternionic 2-dimensional vector subspace W ⊂ ℍ . Here we prove: 16 2 Theorem 1.3 The oriented 4-planes of ℝ ≅ 𝕆 calibrated by the 4-form Φ are the Spin(8) transversal Cayley 4-planes, i.e. the 4-planes P such that both projections (P) ,  (P) on 2 � the two summands in  =  ⊕  are two dimensional and both invariant by a same com- plex structure u ∈ S ⊂ Im. Also, by recalling that  decomposes in the union of octonionic lines = {(x, mx), x ∈ , m ∈  ∪ ∞}, meeting pairwise only at (0, 0)∈  , we can state: Theorem  1.4 The oriented 4-planes of ℝ calibrated by the 4-form Φ are the Spin(7)U(1) ones that are invariant under a complex structure u ∈ S ⊂ Im and that are contained in an octonionic line  ⊂  ⊕  , where only m ∈ ℝ and m =∞ are allowed. Thus they are “Cayley 4-planes”, contained in the oriented 8-planes that are the mentioned octonionic lines with m ∈ ℝ ∪∞. 2 Preliminaries The multiplication in the algebra  of octonions can be defined from the one in quaternions � � � ℍ through the Cayley-Dickson process: if x = h + h e, x = h + h e ∈  , then 1 2 1 2 1 3 104 K. B. M. Boydon, P. Piccinni � � � � � ̄ ̄ xx =(h h − h h )+(h h + h h )e, 1 2 2 1 1 2 1 2 ̄ ̄ where product of quaternions is used on the right hand side and h , h are the conjugates 1 2 � � of h , h ∈ ℍ . Like for quaternions, the conjugation x ̄ = h − h e in  relates with the non- 1 2 1 2 � � �� � �� � �� commutativity: xx = x ̄ x ̄ . One has also the associator [x, x , x ]=(xx )x − x(x x ) , that � �� vanishes whenever two among x, x , x ∈  are equal or conjugate. The identification x = h + h e ∈ 𝕆 ↔ (h , h )∈ ℍ , used in the previous formula, is not 1 2 1 2 an isomorphism of (left or right) quaternionic vector spaces. To get an isomorphism one has instead to go through the following hypercomplex structure (I,  J,  K) on ℝ ≅ 𝕆 . For x = h + h e ∈  , where (h , h )∈ ℍ , define 1 2 1 2 I(x)= x ⋅ i, J(x)= x ⋅ j, K(x) =(x ⋅ i) ⋅ j or equivalently I(h , h )=(h i, −h i), J(h , h ))=(h j, −h j), K(x) =(h k, h k). 1 2 1 2 1 2 1 2 1 2 This observation goes likely back to the very discovery of octonions in the mid-1800s. The alternative approach to the same isomorphism used in our Introduction does not seem however to have appeared before 1989, when R. Bryant and R. Harvey [6] looked at the map L ∶ ℍ → 𝕆, L(h , h )= h +(kh k)e ∈ 𝕆 , 1 2 1 2 and observed it satisfies ̄ ̄ L[(h , h )i]= h i +(kh ik)e, L[(h , h )j]= h j +(kh jk)e, L[(h , h )k]= h k +(kh )e. 1 2 1 2 1 2 1 2 1 2 1 2 x = h , x = kh k x = x + x e This, in terms of and of the octonion , can be read exactly 1 1 2 2 1 2 as in our previous approach: L[(h , h )i]= x ⋅ i, L[(h , h )j]= x ⋅ j, L[(h , h )k] =(x ⋅ i) ⋅ j, 1 2 1 2 1 2 and as mentioned L Φ =Φ . Spin(7) K 3 The quaternionic 4‑form and the Cayley calibration in  ℝ A possible way to produce 4-forms canonically associated with some G-structures is through the notion of Clifford system. We recall the definition, originally given in the con- text of isoparametric hypersurfaces, cf. [9]. Definition 3.1 A Clifford system on a Riemannian manifold (M,  g) is a vector sub-bundle E ⊂ End TM locally spanned by self-adjoint anti-commuting involutions I ,… , I . Thus 1 r 2 ∗ I = Id, I = I , I ◦I =−I ◦I , and the I are required to be related, in the inter- sections of trivializing sets, by matrices of SO(r) . The rank r of E is said to be the rank of the Clifford system. Possible ranks of irreducible Clifford systems on ℝ are classified, up to N = 32 , in Table 1. 1 3 Clifford systems, Clifford structures, and their canonical… 105 In particular, the Clifford system of rank 3 in ℝ can be defined by the classical Pauli matrices: 01 0 − i 10 I = , I = , I = ∈ U(2) ⊂ SO(4), 1 2 3 10 i 0 0 − 1 and the Clifford system of rank 5 in ℝ by the following similar (right) quaternionic Pauli matrices: 0 Id 0 − R 0 − R i j I = , I = , I = , 1 2 3 Id 0 R 0 R 0 i j (3.1) 0 − R Id 0 I = , I = ∈ Sp(2) ⊂ SO(8), 4 5 R 0 0 − Id 2 8 where as before R , R , R denote the multiplication on the right by i, j, k on ℍ ≅ ℝ . i j k According to Table 1, there is also a Clifford system with r = 4 in ℝ , explicitly defined by selecting e.g. 0 Id 0 − R 0 − R 0 − R i j k I = , I = , I = , I = . 1 2 3 4 Id 0 R 0 R 0 R 0 i j k Going back to rank r = 5 , from the quaternionic Pauli matrices I , I , I , I , I , one gets 1 2 3 4 5 the 10 complex structures on ℝ I = I ◦I for 1 ≤ 𝛼 <𝛽 ≤ 5. 𝛼𝛽 𝛼 𝛽 Their Kähler forms give rise to a 5 × 5 skew-symmetric matrix =( ), and one can easily see that both the following matrices of Kähler 2-forms ⎛ ⎞ L L i j ⎜ ⎟ =( )∈ (5) and  = − 0  ∈ (3) R  L L L i k ⎜ ⎟ − −  0 ⎝ ⎠ L L j k allow to write the (left) quaternionic 4-form of ℍ as 2 2 2 2 Ω =− 𝜃 =[𝜔 + 𝜔 + 𝜔 ]. 𝛼𝛽 L L L (3.2) i j k 𝛼<𝛽 On the other hand, as mentioned in the Introduction, the subgroup Spin(7) ⊂ SO(8) (generated by the right translation R , u ∈ S ⊂ Im ) gives rise to the Cayley calibration Φ ∈Λ : Spin(7) Table 1 Rank of irreducible Dimension N 2 4 8 8 16 16 16 16 32 64 64 ... Clifford systems in ℝ Rank r 2 3 4 5 6 7 8 9 10 11 12 ... 1 3 106 K. B. M. Boydon, P. Piccinni 1 1 2 2 2 2 Φ =− [𝜙 + 𝜙 +… 𝜙 ]= 𝜁 . Spin(7) i j h 𝛼𝛽 6 6 𝛼<𝛽 Here  ,  ,  ,  ,  ,  ,  . are the Kähler 2-forms associated with the complex struc- i j k e f g h tures (J , J , J , J , J , J , J )= (R , R , R , R , R , R , R ) , and  =( )∈ (7) is the 1 2 3 4 5 6 7 i j k e f g h matrix of the Kähler 2-forms of compositions J = J ◦J . It is worth to recall that under the action of Sp(2)Sp(1), the space of exterior 2-forms 2 8 Λ ℝ decomposes as 2 2 2 2 Λ =Λ ⊕ Λ ⊕ Λ , 10 15 3 (cf. [22, page 125] as well as [23, page 93]). Here lower indices denote the dimensions of irreducible components. Here Λ ≅ (2) is generated by the Kähler forms  of the J (𝛼 <𝛽 ) , compositions of the five quaternionic Pauli matrices, and Λ ≅ (1) is gener- 𝛼𝛽 ated by the Kähler forms  , , . L L L i j k By denoting by  the second coefficient in the characteristic polynomial of the involved skew-symmetric matrices, we can rewrite formula (3.2) of Ω as: Ω =−  ( )=  ( ), L 2 R 2 L ⎛ L L ⎞ i j ⎜ ⎟ = − 0  ∈ (3) where  =( )∈ (5) , and . L L R  L i k ⎜ ⎟ ⎝− −  0 ⎠ L L j k Similarly, under the Spin(7) action one gets the decomposition: 2 2 2 Λ =Λ ⊕ Λ , 7 21 see [13, page 256] or [14, page 47]. Here Λ is generated by the Kähler forms  of J = R , R ,… , R and Λ ≅ (7) is generated by the Kähler forms  of the the i j h J ◦J (𝛼 <𝛽 ) . Thus, in the  notation: 𝛼 𝛽 2 1 1 Φ =−  =  (),  =( )∈ (7). Spin(7) 2 6 6 All the exterior 4-forms ,  and the have been studied systematically as calibra- tions in the space ℝ , cf. [8]. 4 Proof of Theorems 1.1 and 1.2 The matrix  =( )∈ (5) in the statement of Theorem  1.2 is defined as follows. Let I ( = 1, … ,5) be the left quaternionic Pauli matrices defined as in (3.1) but by using the L L L I = I ◦I left quaternionic multiplications L , L , L by i, j.k. If and if  are the Kähler i j k 2-forms associated to I , a computation shows that 2 2 2 2 2 2 2 2 2 2 2Ω =  +  +  +  +  +  +  +  +  +  , 12 13 24 34 23 14 15 25 35 45 and note the symmetry with the first identity in formula (3.2). We express now the 2-forms  and  in the coordinates of ℝ , using the following 1 8 8 {dx , … , dx } ⊂ Λ ℝ be the standard basis of 1-forms in ℝ . Then abridged notations. Let 1 8 1 3 Clifford systems, Clifford structures, and their canonical… 107 (scriptsize) denotes dx ∧ dx and  denotes dx ∧ dx ∧ dx ∧ dx , and ⋆ denotes the Hodge star, so that a + ⋆ = a + ⋆a . One gets: =−12 + 34 + 56 − 78, =−13 − 24 + 57 + 68, =−14 + 23 + 58 − 67, (4.1) =−14 + 23 − 58 + 67, =+13 + 24 + 57 + 68, =−12 + 34 − 56 + 78, and =−15 − 26 − 37 − 48, =−16 + 25 + 38 − 47, (4.2) =−17 − 28 + 35 + 46, =−18 + 27 − 36 + 45, so that, if  =( ), 2 2 2 𝜏 (𝜃 )= 𝜃 + 𝜃 +⋯ + 𝜃 12 13 45 (4.3) =−121234 − 41256 − 41357 + 41368 − 41278 − 41467 − 41458 + ⋆ =−2Ω . Next: =−12 − 34 + 56 + 78, =−13 + 24 + 57 − 68, =−14 − 23 + 58 + 67, =+14 + 23 + 58 + 67, =−13 + 24 − 57 + 68, (4.4) =+12 + 34 + 56 + 78, =−15 − 26 − 37 − 48, =−16 + 25 − 38 + 47, =−17 + 28 + 35 − 46, =−18 − 27 + 36 + 45, and, if  =( ), 2 2 2 𝜏 (𝜂 )= 𝜂 + 𝜂 +⋯ + 𝜂 12 13 45 (4.5) = 121234 − 41256 − 41357 − 41368 + 41278 + 41467 − 41458 + ⋆ =−2Ω . Similarly: 1 3 108 K. B. M. Boydon, P. Piccinni =−12 + 34 + 56 − 78, =−13 − 24 + 57 + 68, =−14 + 23 + 58 − 67, =−15 − 26 − 37 − 48, e (4.6) =−16 + 25 − 38 + 47, =−17 + 28 + 35 − 46, =−18 − 27 + 36 + 45, and one easily deduces also formulas for the  (cf. [3, 19]). Then, by (1.2): Φ = 1234 + 1256 + 1357 + 1368 − 1278 − 1467 + 1458 + ⋆. Spin(7) (4.7) By computing the squares of the 2-forms in (4.6) (4.4) and comparing with Formulas (4.5), (4.3), (4.7), the identities listed in Theorems 1.1 and 1.2 are recognized. 5 Even Clifford structures in dimension 8 We recall first the following notion, proposed in 2011  by A. Moroianu and U. Semmel- mann, [17]. Definition 5.1 Let (M,  g) be a Riemannian manifold. An even Clifford structure is the choice of an oriented Euclidean vector bundle E of rank r ≥ 2 over M, together with a bundle morphism  from the even Clifford algebra bundle even r 2 r − ∶ Cl E → EndTM such that Λ E ↪ End TM. r is called the rank  of the even Clifford structure. The even Clifford structure E is said to be parallel if there exists a metric connection ∇ on E such that  is connection preserving, i.e. (∇ )= ∇ (), X X even g for every tangent vector X ∈ TM and section  of Cl E , where ∇ is the Levi Civita connection. Rank 2, 3, 4 parallel even Clifford structures are equivalent to complex Kähler, quater - nion Kähler, product of two quaternion Kähler. Besides them, higher rank parallel non-flat even Clifford structures in dimension 8 are listed in Table 2, cf. [17]. A class of examples of even Clifford structures are those coming from Clifford systems as defined in Sect.  3. Namely, if the vector sub-bundle E ⊂ End TM , locally spanned by self-adjoint anticommuting involutions I ,… , I , defines the Clifford system, then 1 r one easily recognizes that through the compositions I = I ◦I , the Clifford morphism even ∶ Cl (E ) → End TM is well defined. An example is given by the first row of Table  2, where the quaternion Kähler structure is constructed via the local I = I ◦I defined as in Sect.  3, by using on the model space ℍ the quaternionic Pauli matrices. The remaining three rows of Table  2  correspond to 1 3 Clifford systems, Clifford structures, and their canonical… 109 Table 2 Parallel non-flat even r M Clifford structures of rank ≥ 5 in M 5 Quaternion Kähler 6 Kähler Spin(7) 7 holonomy 8 Riemannian Table 3 Generators and associated 4-forms r M Clifford bundle generators Associated 4-form 1 1 5 qK I , I , I , I , I 1 2 3 4 5 Ω =−  ( ), Ω =−  ( ) L 2  R 2 2 2 6 Kähler J , J , J , J , J , J Φ =  ( )=−5 1 2 3 4 5 6 Spin(6) 2 1 1 7 Spin(7) hol J , J , J , J , J , J , J 1 2 3 4 5 6 7 Φ =−  =  ( ) Spin(7) 2 6 6 8 Riemannian I, J , J , J , J , J , J , J Φ =  ( )= 0 1 2 3 4 5 6 7 SO(8) 2 essential even Clifford structures, i. e. to even Clifford structures that cannot be defined throw a Clifford system, cf. [21] for a discussion on this notion. Table  3 gives a description of the Clifford bundle generators and of the canonically associated 4-form for each of the four even Clifford structures on ℝ . Here I , I , I , I , I are the (left or right) quaternionic Pauli matrices, ( ) , ( ) are 1 2 3 4 5 like in Sect.  4. Notations (J , J , J , J , J , J , J )= (R , R , R , R , R , R , R ) are also used, 1 2 3 4 5 6 7 i j k e f g h is the Kähler form of J and is the Kähler form of J ◦J . Finally , ( )∈ (8) , with entries ± in the first line and column and with entries  ∈ (7). It is of course desirable to give examples of Riemannian manifolds (M , g) supporting both a Sp(2) ⋅ Sp(1) and a Spin(7) structure. Rarely the metric g can be the same for both structures, but this is possible of course for parallelizable (M , g) . On this respect, homoge- neous (M , g) with an invariant Spin(7) structure have been recently classified [1], by mak - ing use of the following topological condition for compact oriented spin M : p (M)− 4p (M)+ 8(M)= 0. 7 1 Some of the obtained examples are parallelizable, e. g. diffeomorphic to S × S and 5 3 5 3 S × S . On the latter, S × S , using two natural parallelizations, one can define two Spin(7) structures, both of general type (in the 1986 M. Fernández Spin(7) framework), and the hyperhermitian structure associated with one of them corresponds to a family of Calabi- Eckmann [20]. To get examples of 8-dimensional manifolds that admit both a locally conformally hyperkähler metric g and a locally conformal parallel Spin(7) metric, that is either the same g as before, or a different metric g , a good point to start with is the class of compact 7 7 3-Sasakian 7-dimensional manifolds (S , g) . Many examples of such (S , g) and with arbi- trary second Betti numbers have been given by Ch. Boyer, K. Galicki et al, cf. [5]. In par- ticular, recall that given the 3-Sasakian (S , g) one gets a locally conformally hyperkähler 7 1 metric g on the product S × S [18]. This can also be expressed by saying that the 3-Sasa- kian metric g has the property of being nearly parallel G , and in particular with 3 linearly independent Killing spinors, cf. [4, pp. 536–538]. Moreover the differentiable manifold S 1 3 110 K. B. M. Boydon, P. Piccinni admits, besides the 3-Sasakian metric g, another metric g that is also nearly parallel G but proper, i. e. with only one non zero Killing spinor. This allows to extend the metrics g and g to the product with S and to get both the properties of locally conformally hyperkähler 8 7 1 and locally conformally parallel Spin(7) on (M , g)=(S × S , g) and of locally confor- 8 � 1 � mally parallel Spin(7) on (M , g )=(S × S , g ) , cf. also [12]. Further examples of 8-dimensional differentiable manifolds admitting both a Sp(2) ⋅ Sp(1)-structure with respect to a metric g and a Spin(7)-structure with respect to a metric g include the Wolf spaces ℍP and G ∕SO(4) , cf. [1]. Finally, the non singular sex- 5 6 6 tic Y = {[z , … , z ]∈ ℂP , z +⋯ + z = 0} is also an example, where a metric g giving 0 5 0 5 an almost quaternionic structure is insured by a result in [7], and a metric g with holonomy SU(4) ⊂ Spin(7) by Calabi-Yau theorem, cf. [13, p. 139]. 6 Dimension 16 2 16 A Clifford system with r = 9 in 𝕆 ≅ ℝ is given by the following octonionic Pauli matrices: 0 Id 0 − R 0 − R i j I = , I = , I = , 1 2 3 Id 0 R 0 R 0 i j 0 − R 0 − R 0 − R k e f I = , I = , I = , 4 5 6 R 0 R 0 R 0 k e f 0 − R 0 − R Id 0 g h I = , I = , I = ∈ SO(16), 7 8 9 R 0 R 0 0 − Id g h and of course now R , R ,… , R denote the multiplication on the right by the unit octonions i j h 2 16 i, j,… , h on 𝕆 ≅ ℝ . Looking back at Table 1, we see that in ℝ there are also irreducible Clifford systems with r = 8, 7, 6 . According to [21], convenient choices are the following: r = 8 ∶ I ,… , I , r = 7 ∶ I ,… , I , r = 6 ∶ I , I , I , I , I , I . 1 8 2 8 1 2 3 4 5 9 It is now worth to remind the following parallel situations in complex, quaternionic and octonionic geometry. The groups , Sp(2) ⋅ Sp(1) ⊂ SO(8) , Spin(9) ⊂ SO(16) U(2) ⊂ SO(4) are the stabilizers of the vector subspaces 3 + 4 5 + 8 9 + 16 E ⊂ End (ℝ ), E ⊂ End (ℝ ), E ⊂ End (ℝ ) spanned respectively by the Pauli, quaternionic Pauli, octonionic Pauli matrices. Moreover, U(2) , Sp(2) ⋅ Sp(1) , Spin(9) are symmetry groups of the Hopf fibrations respectively: 1 3 7 S S S 3 2 1 7 4 1 15 8 1 S ⟶ S ≅ ℂP , S ⟶ S ≅ ℍP , S ⟶ S ≅ 𝕆 P . 2 2 4 2 8 2 Finally, U(2) , Sp(2) ⋅ Sp(1) , Spin(9) are stabilizers in Λ ℂ ,Λ ℍ ,Λ 𝕆 of the following canonically associated forms, cf. [2]: ∗ 2 ∗ 4 ∗ 8 Φ = p  d𝓁 ∈Λ , Φ = p  d𝓁 ∈Λ , Φ = p  d𝓁 ∈Λ , U(2) 𝓁 Sp(2)⋅Sp(1) 𝓁 Spin(9) 𝓁 𝓁 𝓁 𝓁 1 1 1 ℂP ℍP 𝕆 P 1 3 Clifford systems, Clifford structures, and their canonical… 111 def def 2 2 2 where  is the volume form on the line or in ℂ or ℍ or  , = {(0, y)} = {(x, mx)} 2 4 2 8 2 16 p ∶ ℂ ≅ ℝ or ℍ ≅ ℝ or 𝕆 ≅ ℝ ⟶ is the projection on the line ) , and note that the integral formula is based on the volume of distinguished planes. In the three cases one gets in this way the Kähler 2-form of ℂ , the 2 2 quaternion Kähler 4-form of ℍ and the canonical 8-form of  . 7 Rank 8, 7 and 6 Clifford systems on  ℝ Look now closer at the nine octonionic Pauli matrices, that define a rank 9 Clifford system in ℝ , and at the choices among them that give rise to ranks r = 8, 7, 6 (cf. previous Sec- tion). The compositions I = I ◦I , 𝛼 <𝛽 , for all choices r = 6, 7, 8, 9 are bases of the 𝛼𝛽 𝛼 𝛽 Lie algebras (6) ⊂ (7) ⊂ (8) ⊂ (9) ⊂ (16). Like in the previous Sections, we can write the matrices of Kähler forms  associated to I , and we use for them the following notations: A B C D =( )∈ (6),  =( )∈ (7),  =( )∈ (8),  =( )∈ (9). The second coefficients  of their characteristic polynomial give rise to the following invariant 4-forms A B C D 4 16 ( ),  ( ),  ( ),  ( )∈ Λ ℝ 2 2 2 2 that can be written in (the differentials of) the coordinates of ℝ = 𝕆 ⊕ 𝕆 : 1,2,3,4,5,6,7,8;1 ,2 ,3 ,4 ,5 ,6 ,7 ,8 . We recall in particular that in the Spin(9) situation, the following identity holds: ( )= 0, D 8 and this gives evidence to the next coefficient  ( )∈Λ , proportional to the 8-form Φ , as studied in [19]. Spin(9) 8 The 4‑forms 8 and 8 Spin(8) Spin(7)U(1) Look now only at I ,… , I and at the matrix 1 8 =( )∈ (8) of Kähler forms associated to I = I ◦I . By using coordinate 1-forms 1,2,3,4,5,6,7,8;1 ,2 ,3 ,4 ,5 ,6 ,7 ,8 , an explicit computation, based on the formulas in [19] for the  , yields: 1 3 112 K. B. M. Boydon, P. Piccinni 1 1 2 Φ = 𝜏 (𝜓 )= 𝜓 = 1234 + 1256 − 1278 + 1357 + 1368 + 1458 Spin(8) 2 𝛼𝛽 4 4 1=𝛼<𝛽 4 16 � � − 1467 − 2358 + 2367 + 2457 + 2468 − 3456 + 3478 + 5678 + aba b ∈Λ ℝ . 1=a<b Here boldface notations have the following meaning: � � � � � � � � � � � � abcd = abc d − ab cd + ab c d + a bcd − a bc d + a b cd. By excluding now the Kähler forms involving I and I , one gets the matrix 1 9 =( )∈ (7) . Now similar computations lead to: Φ = 𝜏 (𝜓 )= 6[1234 + 1256 Spin(7)U(1) 2 − 1278 + 1357 + 1368 + 1458 − 1467 − 2358 + 2367 + 2457 + 2468 − 3456 + 3478 + 5678] � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � + 6[1 2 3 4 + 1 2 5 6 − 1 2 7 8 + 1 3 5 7 + 1 3 6 8 + 1 4 5 8 − 1 4 6 7 − 2 3 5 8 � � � � � � � � � � � � � � � � � � � � � � � � � + 2 3 6 7 + 2 4 5 7 + 2 4 6 8 − 3 4 5 6 + 3 4 7 8 + 5 6 7 8]+ 6 aba b 1=a<b + 2[1234 + 1256 − 1278 + 1357 + 1368 + 1458 − 1467 − 2358 4 16 + 2367 + 2457 + 2468 − 3456 + 3478 + 5678] ∈Λ ℝ , where boldface notations have the same meaning as before. The presence of the factor U(1) in the group Spin(7)U(1) is here due to a computation showing that matrices in SO(16) commuting with the seven involutions I ,… I are a U(1) subgroup, well identified in [3, 2 8 Chapter 6, p. 44] (cf. also proof of Theorem 1.4 below). We take this opportunity to remark that the expression of Φ written in the paper Spin(7)U(1) [21] contains some errors in the coefficients as well as some missing terms, and has to be corrected by the present one. Note also that Φ restricts, on any of the two sum- Spin(7)U(1) 16 8 8 mands of ℝ = ℝ ⊕ ℝ , and up to a factor 6, to the usual Cayley calibration of [11]. We are now ready for the proofs of Theorems 1.3 and 1.4. The following notion has already implicitly introduced in the statement of Theorem 1.3. 16 2 � Definition 8.1 Let P be a 4-plane in the real vector space ℝ ≅ 𝕆 = 𝕆 ⊕ 𝕆 , and let 2 � 2 � ∶  →  and  ∶  →  be the orthogonal projections to  and  . P is said to be a transversal Cayley 4-plane if both its projections (P) ,  (P) are 2-dimensional and invari- ant under a same complex structure u ∈ S ⊂ Im. Proof of Theorem 1.3 Recall that Spin(8) can be characterized as the subgroup of the fol- lowing matrices A ∈ SO(16): a 0 A = 0 a where a , a ∈ SO(8) are triality companions, i. e. and for any v ∈  there exists a w ∈ + − such that R = a R a (cf. [10, p. 278-279]). It follows that Spin(8) contains the diagonal w + v Spin(7) (characterized by choices a = a ) and acts transitively on transversal 4-planes Δ + − of ℝ . On the other hand the 4-form Φ is invariant under the action of Spin(8) . Thus, Spin(8) since Φ takes value 1 on the 4-plane spanned by the coordinates 121 2 , Φ takes Spin(8) Spin(8) value 1 on any tranversal Cayley 4-plane in ℝ . ◻ 1 3 Clifford systems, Clifford structures, and their canonical… 113 Next, let Q be any 4-plane of ℝ . By looking at the expression of Φ , we see that Spin(8) the only possibilities for having non zero value on Q are that (Q) and  (Q) are 2-dimen- sional. For such 4-planes Q we can use the following canonical form with respect to the complex structure i ∈ S : � � � � � Q = e ∧(R e cos 𝜃 + e sin 𝜃 ) ⊕ e ∧(R e cos 𝜃 + e sin 𝜃 ) , 1 i 1 2 i 1 1 2 where the pairs e , e and e , e are both orthonormal and respectively in  and in  , and 1 2 1 2 0 ≤  ≤ with angles limited by and  ≤  ≤  −  . The above canonical form for Q is a small variation of the canonical forms that are used in a proof of the classical Wirtinger’s inequaliy (cf. [16, p.6]) and in characterizations of Cayley 4-planes in ℝ in the Harvey- Lawson foundational paper (cf. [11, p. 121]). Its proof follows the steps of proof of the mentioned canonical form, as explained in details in [16]. From this canonical form we see that Φ (Q) ≤ 1 for any 4-plane Q, and that the equality holds only if  =  = 0 , i. e. Spin(8) for transversal Cayley 4-planes. Proof of Theorem 1.4 The leading terms in the expression of Φ are those with coef- Spin(7)U(1) ficient 6, thus terms involving only coordinates among 12345678 , or only coordinates among 1 2 3 4 5 6 7 8 , or terms aba b . Look first at the first and second types of terms. We already 2 � =  ⊕ mentioned that the restriction of Φ to any of the summands in is Spin(7)U(1) the usual Cayley calibration in ℝ , whose calibrated 4-planes are the Cayley planes. Thus, for the first two types of terms, we get as calibrated 4-planes just the Cayley 4-planes that are contained in the octonionic lines with slope m = 0 and m =∞ . In the remaining case of terms aba b one gets as calibrated 4-planes the transversal Cayley 4-planes that are con- tained in the octonionic line  (leading coefficient m = 1 ). Now Spin(7) acts on the indi- ,  , vidual octonionic lines , and the only possibility to move planes out of them is 0 1 ∞ through the factor U(1) . In fact, the discussion in [3, Chapter 6, p. 44] shows that the factor U(1) in the group Spin(7)U(1) moves the octonionic lines through the circle, contained in the space S of the octonionic lines, passing through the three points m = 0, 1,∞ . This cor- responds to admitting any real coefficient: m ∈ ℝ ∪∞ as slope of the octonionic lines that are admitted to contain the calibrated 4-planes. ◻ Remark 8.2 Following the recent work [15] by J. Kotrbatý, one can use octonionic 1-forms, according to the following formal definitions: dx = d + id + jd + kd + ed + fd + gd + hd, dx = d − id − jd − kd − ed − fd − gd − hd, � � � � � � � � � dx = d + id + jd + kd + ed + fd + gd + hd , � � � � � � � � dx = d − id − jd − kd − ed − fd − gd − hd , � 16 referring to pairs of octonions (x, x )∈ 𝕆 ⊕ 𝕆 = ℝ . Then, in the same spirit proposed in [15], a straightforward computation yields the following formula, much simpler way to write the Spin(8) canonical 4-form of ℝ : Φ = (dx ∧ dx )∧(dx ∧ dx). Spin(8) Similarly, one gets that the Spin(7)U(1) canonical 4-form of ℝ can be written in octon- ionic 1-forms as: 1 3 114 K. B. M. Boydon, P. Piccinni 2 � 2 Φ = (dx ∧ dx) +(dx ∧ dx ) Spin(7)U(1) � 2 2 � � � − (dx ∧ dx ) +(dx ∧ dx) − (dx ∧ dx )∧(dx ∧ dx) . Details of both computations are in [3]. Acknowledgements Kai Brynne M. Boydon was supported by University of the Philippines OVPAA Doc- toral Fellowship. Part of the present work was done during her visit at Sapienza Università di Roma in the academic year 2018–19, and she thanks Sapienza University and Department of Mathematics “Guido Castelnuovo” for hospitality. Paolo Piccinni was supported by the group GNSAGA of INdAM, by the PRIN Project of MIUR “Real and Complex Manifolds: Topology, Geometry and holomorphic dynamics”, and by Sapienza Università di Roma Project “Polynomial identities and combinatorial methods in algebraic and geometric structures”. Funding Open access funding provided by Università degli Studi di Roma La Sapienza within the CRUI- CARE Agreement. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Com- mons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creat iveco mmons .org/licen ses/by/4.0/. References 1. 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Journal

Abhandlungen aus dem Mathematischen Seminar der Universität HamburgSpringer Journals

Published: Apr 1, 2021

Keywords: Octonions; Clifford system; Clifford structure; Calibration; Canonical form; Primary 53C26; 53C27; 53C38

References