Generality of the quaternionic contact structures

Transcription

1 Generality of the quaternionic contact structures Jan Slovák Masaryk University, Brno, Czech Republic joint work with Ivan Minchev, Brno / Sofia Warsaw, November 16, 2017

2 1 Our motivation 2 Geometric structures as EDS 3 The qc structures 4 The Cartan test for qc structures

3 1 Our motivation 2 Geometric structures as EDS 3 The qc structures 4 The Cartan test for qc structures

4 Claude LeBrun, Olivier Biquard In 1991, LeBrun exploited Salamon s twistor correspondence and found a large class of metrics on R 4n+4 with special holonomy Sp(1)Sp(n), cf. his paper On complete quaterninonic-kähler manifolds in Duke. He interpreted the disk B 4n+4 Sp(n + 1, 1)/Sp(1)Sp(n) and his class was parameterized by one holomorphic function on the twistor space Z CP ( 2n + 3).

5 Claude LeBrun, Olivier Biquard In 1991, LeBrun exploited Salamon s twistor correspondence and found a large class of metrics on R 4n+4 with special holonomy Sp(1)Sp(n), cf. his paper On complete quaterninonic-kähler manifolds in Duke. He interpreted the disk B 4n+4 Sp(n + 1, 1)/Sp(1)Sp(n) and his class was parameterized by one holomorphic function on the twistor space Z CP ( 2n + 3). A bit later, Biquard desribed the induced structures, the so called quaternionic contact structures on the boundary more carefully, cf. Métriques d Einstein asymptotiquement symétriques, Astérisque 265 (2000). He also showed that each real analytic qc structure on a manifold M is the conformal boundary at innity of a (germ) unique quaternionic Kähler metric deffined in a small neighborhood of M. In particular, it was immediately known that there are large families of examples there.

6 Robert Bryant Srní 2015 In the three lectures G-structures with prescribed symmetry at the Geometry and Physics in Srni, January 2015, Bryant delivered a detailed exposition how to use the Cartan-Kähler theory when describing the generality of geometric structures. See for the full texts of the lectures.

7 Robert Bryant Srní 2015 In the three lectures G-structures with prescribed symmetry at the Geometry and Physics in Srni, January 2015, Bryant delivered a detailed exposition how to use the Cartan-Kähler theory when describing the generality of geometric structures. See for the full texts of the lectures. Ivan Minchev thought one could proceed this way in the case of the qc-structures. This meant, he wanted to recover completely the Chern-Moser approach to CR-structures in order to get the necessary data for the Cartan s test. I will describe the result of this quest.

8 1 Our motivation 2 Geometric structures as EDS 3 The qc structures 4 The Cartan test for qc structures

9 We adopt the following ranges of indices: 1 a, b, c, d, e n, 1 s l, where l and n are some fixed positive integers. Problem: Given a set of real analytic functions C a bc : Rl R with C a bc = C a cb, find linearly independent one-forms ωa, defined on a domain Ω R n, and a mapping u = (u s ) : Ω R l so that the equations are satified everywhere on Ω. dω a = 1 2 C a bc (u) ωb ω c (1)

10 We adopt the following ranges of indices: 1 a, b, c, d, e n, 1 s l, where l and n are some fixed positive integers. Problem: Given a set of real analytic functions C a bc : Rl R with C a bc = C a cb, find linearly independent one-forms ωa, defined on a domain Ω R n, and a mapping u = (u s ) : Ω R l so that the equations dω a = 1 2 C a bc (u) ωb ω c (1) are satified everywhere on Ω. The problem is diffeomorphism invariant in the sense that if (ω a, u) is any solution of (1) defined on Ω R n and Φ : Ω Ω is a diffeomorphism, then (Φ (ω a ), Φ (u)) is a solution of (1) on Ω. We regard any such two solutions as equivalent and we are interested in the following question:

11 We adopt the following ranges of indices: 1 a, b, c, d, e n, 1 s l, where l and n are some fixed positive integers. Problem: Given a set of real analytic functions C a bc : Rl R with C a bc = C a cb, find linearly independent one-forms ωa, defined on a domain Ω R n, and a mapping u = (u s ) : Ω R l so that the equations dω a = 1 2 C a bc (u) ωb ω c (1) are satified everywhere on Ω. The problem is diffeomorphism invariant in the sense that if (ω a, u) is any solution of (1) defined on Ω R n and Φ : Ω Ω is a diffeomorphism, then (Φ (ω a ), Φ (u)) is a solution of (1) on Ω. We regard any such two solutions as equivalent and we are interested in the following question: How many non-equivalent solutions does a given problem of this type admit?

12 EDS reformulation We set with the projections N = GL(n, R) R n R l p = (p a b ) : N GL(n, R), x = (x a ) : N R n, u = (u s ) : N R l. Further set ω a def = pb a dx b, and consider the differential ideal I on N generated by the set of two-forms Υ a def = dω a C a bc (u) ωb ω c. Then, the solutions of (1) are precisely the n-dimensional integral manifolds of I on which the restriction of the n-form ω 1 ω n is nowhere vanishing.

13 In our equations dω a = 1 2 C a bc (u) ωb ω c, the functions C a bc play the role curvature. Their derivatives are driven by the Bianchi identities and, thus, they are quadratic. In order to employ the Cartan-Kähler theory we need to replace the quadratic terms by some linear objects. Thus we posit the following two assumptions:

14 In our equations dω a = 1 2 C a bc (u) ωb ω c, the functions C a bc play the role curvature. Their derivatives are driven by the Bianchi identities and, thus, they are quadratic. In order to employ the Cartan-Kähler theory we need to replace the quadratic terms by some linear objects. Thus we posit the following two assumptions: Assumption I: Let us assume that there exist a real analytic mapping F = (F s a ) : R l R ln for which d ( C a bc ωb ω c) = C a bc (u) u s ( du s + F s d (u) ωd) ω b ω c. (2)

15 In our equations dω a = 1 2 C a bc (u) ωb ω c, the functions C a bc play the role curvature. Their derivatives are driven by the Bianchi identities and, thus, they are quadratic. In order to employ the Cartan-Kähler theory we need to replace the quadratic terms by some linear objects. Thus we posit the following two assumptions: Assumption I: Let us assume that there exist a real analytic mapping F = (F s a ) : R l R ln for which d ( C a bc ωb ω c) = C a bc (u) u s ( du s + F s d (u) ωd) ω b ω c. (2) Assumption II: On the integral manifolds of I, the RHS of (2) has to vanish, which is a system of algebraic equations for the unknown 1-forms du s (for a fixed u). We assume that it is non-degenerate, i.e, that equations yield du s span{ω a }. As a consequence, at any u, the set of all solutions du s is parameterized by a certain vector space. We will assume that the dimension of this vector space is a constant D (independent of u).

16 I is a differential ideal, thus it is algebraically generated by the forms Υ a and dυ a. By (2), we have 2dΥ a = C a bc (u) u s ( du s + F s d (u)ωd) ω b ω c + 2C a bc Υb ω c (3) and therefore, I is algebraically generated by Υ a and the 3-forms Ξ a def = C a bc (u) u s ( du s + F s d (u)ωd) ω b ω c.

17 I is a differential ideal, thus it is algebraically generated by the forms Υ a and dυ a. By (2), we have 2dΥ a = C a bc (u) u s ( du s + F s d (u)ωd) ω b ω c + 2C a bc Υb ω c (3) and therefore, I is algebraically generated by Υ a and the 3-forms Ξ a def = C bc a (u) ( u s du s + Fd s (u)ωd) ω b ω c. If we take Ω a to be some other basis of one-forms for the vector space span{ω a }, we can express the forms Ξ a as Ξ a = Π a bc Ωb Ω c, (4) where Π a bc one-forms are linear combinations of the linearly independent {du s + F s d (u)ωd : s = 1,..., n}.

18 The Cartan s test Next, define v 1 (u), v 2 (u),..., v n (u), non-negative integers for any fixed u, as follows:

19 The Cartan s test Next, define v 1 (u), v 2 (u),..., v n (u), non-negative integers for any fixed u, as follows: v 1 (u) = 0, } v d (u) = rank {Π a bc (u) : a = 1,..., n, 1 b < c d { } rank Π a bc : a = 1,..., n, 1 b < c d 1 ; for 1 < d n 1, and { } v n (u) = l rank Π a bc : a = 1... n, 1 b < c n 1.

20 The Cartan s test Next, define v 1 (u), v 2 (u),..., v n (u), non-negative integers for any fixed u, as follows: v 1 (u) = 0, } v d (u) = rank {Π a bc (u) : a = 1,..., n, 1 b < c d { } rank Π a bc : a = 1,..., n, 1 b < c d 1 ; for 1 < d n 1, and { } v n (u) = l rank Π a bc : a = 1... n, 1 b < c n 1. If, for every u R l, one can find a basis Ω a of span{ω a } for which the Cartan s Test v 1 (u) + 2v 2 (u) + + nv n (u) = D, (5) is satisfied (remind D is the constant dimension from the above Assuption II), then the system (1) is said to be in involution.

21 The latter method of computation for the Cartan s character sequence of an ideal is based on the big Bryant et al EDS book, Proposition Theorem (Cartan, essentially, see the Bryant et al EDS book) If the system is in involution, then for any u 0, there exists a solution (ω a, u) of (1) defined on a neighborhood Ω of 0 R n for which u(0) = u 0 and du s 0 = F s d (u 0)ω d 0.

22 The latter method of computation for the Cartan s character sequence of an ideal is based on the big Bryant et al EDS book, Proposition Theorem (Cartan, essentially, see the Bryant et al EDS book) If the system is in involution, then for any u 0, there exists a solution (ω a, u) of (1) defined on a neighborhood Ω of 0 R n for which u(0) = u 0 and du s 0 = F s d (u 0)ω d 0. Moreover, in certain sense (see again the EDS book for a more precise formulation): The generality of the solutions Different solutions (ω a, u) of (1), modulo diffeomorphisms, depend on v k (u) functions of k variables, where v k (u) is the last non-vanishing integer in the Cartan s sequence v 1 (u),..., v n (u).

23 1 Our motivation 2 Geometric structures as EDS 3 The qc structures 4 The Cartan test for qc structures

24 The flat model The spheres appear as Klein models G G/P in many ways, e.g. the conformal Riemannian sphere S n RP n+1, the CR-sphere S 2n+1 CP n+1, and the quaternionic contact sphere S 4n+3 HP n+1, respectively, or other nice homogeneous spaces in the cases of other than positive definite signatures. All these geometries appear as boundaries of domains, carrying a lot of information let us mention the conformal horizons in mathematical physics, the boundaries of domains in complex analysis and function theory, and the boundaries of quaternionic-kähler domains.

25 The flat model The spheres appear as Klein models G G/P in many ways, e.g. the conformal Riemannian sphere S n RP n+1, the CR-sphere S 2n+1 CP n+1, and the quaternionic contact sphere S 4n+3 HP n+1, respectively, or other nice homogeneous spaces in the cases of other than positive definite signatures. All these geometries appear as boundaries of domains, carrying a lot of information let us mention the conformal horizons in mathematical physics, the boundaries of domains in complex analysis and function theory, and the boundaries of quaternionic-kähler domains. The regular Cartan geometries modelled on the hypersphere in HP n+1 are called the quaternionic contact geometries.

26 The corresponding Lie algebras enjoy very similar algebraic structures with 2 -gradings where g 0 further splits as h g 0, as indicated symbolically in the matrix (the entries mean those computed from the symmetries of the matrix) h g 1 g 2 g 1 g 0 g 2

27 The corresponding Lie algebras enjoy very similar algebraic structures with 2 -gradings where g 0 further splits as h g 0, as indicated symbolically in the matrix (the entries mean those computed from the symmetries of the matrix) h g 1 g 2 g 1 g 0 g 2 The corresponding Lie algebras g are so(p + 1, q + 1), su(p + 1, q + 1), and sp(p + 1, q + 1). Thus viewing them as matrix algebras over K = R, C, H, they always have columns and rows of width 1, n, 1, respectively, and h = K, g 1 = K n, g 1 = K n, g 2 is the imaginary part of K (thus vanishing in the case K = R) and g 0 is the algebra of the same type as g of signature (p, q).

28 Quaternionic contact geometries Following the work of Biquard, this is a type of geometric structure describing the Carnot-Carathéodory geometry of the boundary at infinity of quaternionic Kähler manifolds. From the point of view of Cartan geometry, this is an instance of parabolic geometries.

29 Quaternionic contact geometries Following the work of Biquard, this is a type of geometric structure describing the Carnot-Carathéodory geometry of the boundary at infinity of quaternionic Kähler manifolds. From the point of view of Cartan geometry, this is an instance of parabolic geometries. Notice, the first cohomology H 1 (g/p, g) appears only in negative homogeneities, thus the entire geometry is completely defined by the distribution T 1 M TM of codimension three. This means that if there were a pre-quaternionic vector space structure on T 1 M for which an algebraic bracket [, ] alg : Λ 2 T 1 M TM/T 1 M would be an imaginary part of a hermitian form, then this structure is essentially unique.

30 Quaternionic contact geometries Following the work of Biquard, this is a type of geometric structure describing the Carnot-Carathéodory geometry of the boundary at infinity of quaternionic Kähler manifolds. From the point of view of Cartan geometry, this is an instance of parabolic geometries. Notice, the first cohomology H 1 (g/p, g) appears only in negative homogeneities, thus the entire geometry is completely defined by the distribution T 1 M TM of codimension three. This means that if there were a pre-quaternionic vector space structure on T 1 M for which an algebraic bracket [, ] alg : Λ 2 T 1 M TM/T 1 M would be an imaginary part of a hermitian form, then this structure is essentially unique. Notice, the latter assumption is quite rigid. For example, on hypersurfaces in H n+1 we obtain the Levi form compatible with the inherited quaternionic structures if and only if the hypersurface is locally isomorphic to the 3-Sasakian sphere S 4n+3.

31 The definition of the geometry Let M be a (4n + 3)-dimensional manifold and H be a smooth distribution on M of codimension three. The pair (M, H) is said to be a quaternionic contact structure if around each point of M there exist 1-forms η 1, η 2, η 3 with the common kernel H, a positive definite inner product g on H, and endomorphisms I 1, I 2, I 3 of H, satisfying (I 1 ) 2 = (I 2 ) 2 = (I 3 ) 2 = id H, I 1 I 2 = I 2 I 1 = I 3, (6) dη s (X, Y ) = 2g(I s X, Y ) for all X, Y H.

32 The definition of the geometry Let M be a (4n + 3)-dimensional manifold and H be a smooth distribution on M of codimension three. The pair (M, H) is said to be a quaternionic contact structure if around each point of M there exist 1-forms η 1, η 2, η 3 with the common kernel H, a positive definite inner product g on H, and endomorphisms I 1, I 2, I 3 of H, satisfying (I 1 ) 2 = (I 2 ) 2 = (I 3 ) 2 = id H, I 1 I 2 = I 2 I 1 = I 3, (6) dη s (X, Y ) = 2g(I s X, Y ) for all X, Y H. As well known, if dim(m) > 7, one can always find, locally, a triple ξ 1, ξ 2, ξ 3 of Reeb vector fields on M satisfying for all X H, η s (ξ t ) = δ s t, η s (ξ t, X ) = dη t (ξ s, X ) (7) (δ s t being the Kronecker delta). In dimension 7, the existence of Reeb vector fields is our additional integrability condition on the qc structure.

33 The Cartan connection (I.M. & J.S., AGAG, 2017) The existence of the unique Cartan connection with co-closed curvature is well known. But in order to apply the Cartan-Kähler theory, we need explicit formulae and knowledge of the structure of the complete curvature and its derivatives.

34 The Cartan connection (I.M. & J.S., AGAG, 2017) The existence of the unique Cartan connection with co-closed curvature is well known. But in order to apply the Cartan-Kähler theory, we need explicit formulae and knowledge of the structure of the complete curvature and its derivatives. The harmonic curvature has got only one component, which is of the type κ H : Λ 2 g 1 g 0. The complete curvature is given as a value of a differential operator on κ H.

35 The Cartan connection (I.M. & J.S., AGAG, 2017) The existence of the unique Cartan connection with co-closed curvature is well known. But in order to apply the Cartan-Kähler theory, we need explicit formulae and knowledge of the structure of the complete curvature and its derivatives. The harmonic curvature has got only one component, which is of the type κ H : Λ 2 g 1 g 0. The complete curvature is given as a value of a differential operator on κ H. By general arguments (due to A. Cap, including BGG machinery), or by direct experiment, it turns out that the curvature cannot have non-trivial values in all the slots indicated by zero: 0 g 1 g 2 0 g 0 0

36 The procedure Consequently, we may proceed in the following steps: Find partial prolongation of the structure equations to the principal fiber bundle P 0 M with the structure algebra H with the normalized structure equations.

37 The procedure Consequently, we may proceed in the following steps: Find partial prolongation of the structure equations to the principal fiber bundle P 0 M with the structure algebra H with the normalized structure equations. Work further on P 0 and construct the complete coframe on the prolongation P 1 P 0 by normalizing the structure equations so that the other two curvature slots vanish, as expected.

38 The procedure Consequently, we may proceed in the following steps: Find partial prolongation of the structure equations to the principal fiber bundle P 0 M with the structure algebra H with the normalized structure equations. Work further on P 0 and construct the complete coframe on the prolongation P 1 P 0 by normalizing the structure equations so that the other two curvature slots vanish, as expected. Observe that the coframe has got the uniqueness and equivariance properties as well as co-closed curvature.

39 The procedure Consequently, we may proceed in the following steps: Find partial prolongation of the structure equations to the principal fiber bundle P 0 M with the structure algebra H with the normalized structure equations. Work further on P 0 and construct the complete coframe on the prolongation P 1 P 0 by normalizing the structure equations so that the other two curvature slots vanish, as expected. Observe that the coframe has got the uniqueness and equivariance properties as well as co-closed curvature. Compute (and parameterize) the differentials of the curvature components explicitly (will be necessary for the Cartan-Kähler approach).

40 The procedure Consequently, we may proceed in the following steps: Find partial prolongation of the structure equations to the principal fiber bundle P 0 M with the structure algebra H with the normalized structure equations. Work further on P 0 and construct the complete coframe on the prolongation P 1 P 0 by normalizing the structure equations so that the other two curvature slots vanish, as expected. Observe that the coframe has got the uniqueness and equivariance properties as well as co-closed curvature. Compute (and parameterize) the differentials of the curvature components explicitly (will be necessary for the Cartan-Kähler approach). (All this is available in the AGAG paper.)

41 Brief account on the computations The small Greek indices α, β, γ,... have got the range 1,..., 2n, whereas the indices s, t, k, l, m are running from 1 to 3. R 4n is considered with its standard inner product, and a quaternionic structure induced by the identification R 4n = H n with the quaternion coordinate space H n, i.e. we fix triple J 1, J 2, J 3 of complex structures which are Hermitian with respect to, and satisfy J 1 J 2 = J 2 J 1 = J 3. The complex vector space C 4n then splits as a direct sum of +i and i eigenspaces, C 4n = W W, with respect to the complex structure J 1. The complex 2-form π, π(u, v) def = J 2 u, v + i J 3 u, v, u, v C 4n, has type (2, 0) with respect to J 1, i.e., it satisfies π(j 1 u, v) = π(u, J 1 v) = iπ(u, v).

42 Further, fix an, -orthonormal basis {e α W, eᾱ W}, eᾱ = e α, with dual basis {e α, eᾱ} so that π = e 1 e n+1 + e 2 e n e n e 2n. Then with, = g α β eα e β + gᾱβ eᾱ e β, π = π αβ e α e β g α β = g βα = { 1, if α = β 0, if α β, π αβ = π βα = 1, if α + n = β 1, if α = β + n 0, otherwise.

43 Further, fix an, -orthonormal basis {e α W, eᾱ W}, eᾱ = e α, with dual basis {e α, eᾱ} so that π = e 1 e n+1 + e 2 e n e n e 2n. Then with, = g α β eα e β + gᾱβ eᾱ e β, π = π αβ e α e β g α β = g βα = { 1, if α = β 0, if α β, π αβ = π βα = 1, if α + n = β 1, if α = β + n 0, otherwise. We also introduce a complex antilinear endomorphism j of tensor algebra of R 4n, which takes a tensor with components T α1...α k β 1... β l... to (jt ) α1...α k β 1... β l... = σ 1... σ k τ 1...τ l... π σ 1 α 1... π σ k α k π τ 1 β 1... π τ l β l... T σ1... σ k τ 1...τ l...

44 The P 0 step If (M, H) is a qc manifold, defined in terms of η s, I s, g, and η 1, η 2, η 3, then there exists a natural principle bundle π 0 : P 0 M with structure group CSO(3) = R + SO(3) whose local sections are precisely the triples of 1-forms (η 1, η 2, η 3 ) satisfying (6). Further, on P 0 we obtain a global triple of canonical one-forms which we denote again by (η 1, η 2, η 3 ). The exterior derivatives of the canonical one-forms are dη 1 = ϕ 0 η 1 ϕ 2 η 3 + ϕ 3 η 2 + 2ig α β θα θ β dη 2 = ϕ 0 η 2 ϕ 3 η 1 + ϕ 1 η 3 + π αβ θ α θ β + πᾱ β θᾱ θ β dη 3 = ϕ 0 η 3 ϕ 1 η 2 + ϕ 2 η 1 iπ αβ θ α θ β + iπᾱ β θᾱ θ β, (8) where ϕ 0, ϕ 1, ϕ 2, ϕ 3 are some (local, non-unique) real one-forms on P 0, θ α are some (local, non-unique) complex and semibasic one-froms on P 0.

45 Towards the P 1 prolongation One can show that, if ϕ 0, ϕ 1, ϕ 2, ϕ 3, θ α are any other one-forms (with the same properties as ϕ 0, ϕ 1, ϕ 2, ϕ 3, θ α ) that satisfy (8), then θ α = Uβ αθβ + ir α η 1 + πσ ᾱ r σ (η 2 + iη 3 ) ϕ 0 = ϕ 0 + 2U β σ r σ θ β + 2U βσ r σ θ β + λ 1 η 1 + λ 2 η 2 + λ 3 η 3 ϕ 1 = ϕ 1 2iU β σ r σ θ β + 2iU βσ r σ θ β + 2r σ r σ η 1 λ 3 η 2 + λ 2 η 3, ϕ 2 = ϕ 2 2π στ Uβ σr τ θ β 2π σ τ U σ β r τ θ β + λ 3 η 1 + 2r σ r σ η 2 λ 1 η 3, ϕ 3 = ϕ 3 + 2iπ στ Uβ σr τ θ β 2iπ σ τ U σ β r τ θ β λ 2 η 1 + λ 1 η 2 + 2r σ r σ η 3, (9) where Uβ α, r α, λ s are some appropriate functions; λ 1, λ 2, λ 3 are real, and {Uβ α} Sp(n) End(R4n ). The functions Uβ α, r α and λ s give a parametrization of a Lie group diffeomorphic to Sp(n) R 4n+3.

46 The P 1 bundle and the canonical coframe There is a canonical principle bundle π 1 : P 1 P 0 whose local sections are the one-forms ϕ 0, ϕ 1, ϕ 2, ϕ 3, θ α on P 0 satisfying (8). We use ϕ 0, ϕ 1, ϕ 2, ϕ 3, θ α to denote also the induced canonical (global) one-forms on the principal bundle P 1. Then, on P 1, there exists a unique set of complex one-forms Γ αβ, φ α and real one-forms ψ 1, ψ 2, ψ 3 so that Γ αβ = Γ βα, (jγ) αβ = Γ αβ. (10) and the following equations hold true: dθ α = iφ α η 1 πσ ᾱφ σ (η 2 + iη 3 ) π ασ Γ σβ θ β 1 2 (ϕ 0 + iϕ 1 ) θ α 1 2 πᾱ(ϕ β 2 + iϕ 3 ) θ β dϕ 0 = ψ 1 η 1 ψ 2 η 2 ψ 3 η 3 2φ β θ β 2φ β θ β dϕ 1 = ϕ 2 ϕ 3 ψ 2 η 3 + ψ 3 η 2 + 2iφ β θ β 2iφ β θ β dϕ 2 = ϕ 3 ϕ 1 ψ 3 η 1 + ψ 1 η 3 2π σβ φ σ θ β 2π σ β φ σ θ β dϕ 3 = ϕ 1 ϕ 2 ψ 1 η 2 + ψ 2 η 1 + 2iπ σβ φ σ θ β 2iπ σ βφ σ θ β, (11)

47 The complete structure equations on P 1 The latter one-forms {η s }, {θ α }, {ϕ 0 }, {ϕ s }, {Γ αβ }, {φ α }, {ψ s } represent the components of the canonical Cartan connection corresponding to a fixed splitting of the relevant Lie algebra sp(n + 1, 1) = g 2 g 1 R sp(1) sp(n) }{{} g 0 g 1 g 2. We have seen big part of the structure equations already, the remaining ones will display the curvatures components satisfying: S αβγδ, V αβγ, L αβ, M αβ, C α, H α, P, Q, R

48 (I) Each of the arrays {S αβγδ }, {V αβγ }, {L αβ }, {M αβ } is totally symmetric in its indices.

49 (I) Each of the arrays {S αβγδ }, {V αβγ }, {L αβ }, {M αβ } is totally symmetric in its indices. (II) We have (js) αβγδ = S αβγδ (jl) αβ = L αβ (12) R = R.

50 (I) Each of the arrays {S αβγδ }, {V αβγ }, {L αβ }, {M αβ } is totally symmetric in its indices. (II) We have (js) αβγδ = S αβγδ (jl) αβ = L αβ (12) R = R. (III) The exterior derivatives of the connection one-forms Γ αβ, φ α and ψ s are given by dγ αβ = π στ Γ ασ Γ τβ + 2πα σ (φ β θ σ φ σ θ β ) + 2πβ σ (φα θ σ φ σ θα) ( + π σ δ S αβγσ θ γ θ δ + V αβγ θ γ + πα σ πβ τ V σ τ γ θ γ) η 1 iπ σ γ V αβσ θ γ (η 2 + iη 3 ) + i(jv) αβγ θ γ (η 2 iη 3 ) il αβ (η 2 + iη 3 ) (η 2 iη 3 ) + M αβ η 1 (η 2 + iη 3 ) + (jm) αβ η 1 (η 2 iη 3 ), (13)

51 dφ α = 1 2 (ϕ 0 + iϕ 1 ) φ α παγ(ϕ 2 iϕ 3 ) φ γ π σ α Γ σ γ φ γ i 2 ψ 1 θ α 1 2 παγ(ψ 2 iψ 3 ) θ γ iπ σ δ Vαγσ θγ θ δ + M αγ θ γ η 1 + π σ α L σ γ θ γ η 1 + il αγ θ γ (η 2 iη 3 ) iπ σ γ M ασ θ γ (η 2 + iη 3 ) C α(η 2 + iη 3 ) (η 2 iη 3 ) + H α η 1 (η 2 + iη 3 ) + iπ ασ C σ η 1 (η 2 iη 3 ), dψ 1 = ϕ 0 ψ 1 ϕ 2 ψ 3 + ϕ 3 ψ 2 4iφ γ φ γ + 4π σ δ Lγσ θγ θ δ + 4C γ θ γ η 1 + 4C γ θ γ η 1 4iπ γ C σ σ θ γ (η 2 + iη 3 ) + 4iπ γσ C σ θ γ (η 2 iη 3 ) + P η 1 (η 2 + iη 3 ) + P η 1 (η 2 iη 3 ) + ir (η 2 + iη 3 ) (η 2 iη 3 ), dψ 2 + i dψ 3 = (ϕ 0 iϕ 1 ) (ψ 2 + iψ 3 ) + i(ϕ 2 + iϕ 3 ) ψ 1 + 4π γδ φ γ φ δ + 4iπγ σ M σ δ θγ θ δ + 4iπγ σ C σ θ γ η 1 4H γ θ γ η 1 4iC γ θ γ (η 2 + iη 3 ) 4iπγ σ H σ θγ (η 2 iη 3 ) ir η 1 (η 2 + iη 3 ) + Q η 1 (η 2 iη 3 ) P (η 2 + iη 3 ) (η 2 iη 3 ).

52 Structure of the curvature All the potentially nonzero curvature components of κ, as just deduced, are listed in the table: homogeneity the cochains object in structure eq. 2 g 1 g 1 sp(n) S αβγδ 3 g 2 g 1 sp(n) V αβγ 3 g 1 g 1 g 1 V αβγ 4 g 2 g 2 sp(n) L αβ, M αβ 4 g 2 g 1 g 1 L αβ, M αβ 4 g 1 g 1 g 2 L αβ, M αβ 5 g 2 g 2 g 1 C α, H α 5 g 2 g 1 g 2 C α, H α 6 g 2 g 2 g 2 P, Q, R Notice that it is the κ = 0 normalization which enforces several potentially different components to coincide.

53 1 Our motivation 2 Geometric structures as EDS 3 The qc structures 4 The Cartan test for qc structures

54 Essentially finished followup paper with I.M. Now we come to the Cartan test quest. In order to show that the Assumptions I and II on the geometric structures via EDS holds true, we may observe that the Bianchi identities imply that the one-forms expressing the differentials of the curvature functions belong to the linear span of η 1, η 2, η 3, θ α, θᾱ (quite lengthy and technical, all already in the AGAG paper).

55 Essentially finished followup paper with I.M. Now we come to the Cartan test quest. In order to show that the Assumptions I and II on the geometric structures via EDS holds true, we may observe that the Bianchi identities imply that the one-forms expressing the differentials of the curvature functions belong to the linear span of η 1, η 2, η 3, θ α, θᾱ (quite lengthy and technical, all already in the AGAG paper). Furthermore, if we are considering the Bianchi identities as a system of algebraic equations for the unknown one-forms expressing the differentials of the curvature functions, then the solutions may be parametrized by elements of a certain vector space.

56 More explicitly, on P 1, there exist unique, globally defined, complex valued functions A αβγδɛ, B αβγδ, C αβγδ, D αβγ, E αβγ, F αβγ, G αβ, X αβ, Y αβ, Z αβ, (N 1 ) α, (N 2 ) α, (N 3 ) α, (N 4 ) α, (N 5 ) α, U s, W s (14) so that: (I) Each of the arrays {A αβγδɛ }, {B αβγδ }, {C αβγδ }, {D αβγ }, {E αβγ }, {F αβγ }, {G αβγ }, {X αβ }, {Y αβ }, {Z αβ } is totally symmetric in its indices. (II) The differentials of all the curvature components are expressed in a linear way by means of the latter tensors and the curvature and the coframe.

57 More explicitly, on P 1, there exist unique, globally defined, complex valued functions A αβγδɛ, B αβγδ, C αβγδ, D αβγ, E αβγ, F αβγ, G αβ, X αβ, Y αβ, Z αβ, (N 1 ) α, (N 2 ) α, (N 3 ) α, (N 4 ) α, (N 5 ) α, U s, W s (14) so that: (I) Each of the arrays {A αβγδɛ }, {B αβγδ }, {C αβγδ }, {D αβγ }, {E αβγ }, {F αβγ }, {G αβγ }, {X αβ }, {Y αβ }, {Z αβ } is totally symmetric in its indices. (II) The differentials of all the curvature components are expressed in a linear way by means of the latter tensors and the curvature and the coframe. This concludes, that the qc structures might be discussed via the Cartan test.

58 For the (real) dimension D of the vector space determined by (14), we calculate ( ) 2n + 4 D = ( 2n ) +6 ( 2n ) +8 ( 2n ) +20n+12 = 2 (2n + 5)(2n + 3)(n + 3)(n + 2)(n + 1). 15 In order to show that our exterior differential system I is in involution which would allow us to apply the Cartan s Third Theorem to it we need to compute the character sequence v 1, v 2, v 3,..., v d1 of the system and show that the Cartan s test D = v 1 + 2v 2 + 3v d 1 v d1 is satisfied, where d 1 = (2n + 5)(n + 2) is the dimension of P 1.

59 A lengthy and technical computation shows (1 λ n) v λ = 1 2 (λ 1)(λ 2n 4)(λ 2n 5) v n+λ = 1 2 (n + λ 1)(n λ + 4)(n λ + 5) = 12n v 2n+1 v 2n+2 = 6n + 3 v 2n+3 = 2n + 2 v 2n+4 = v 2n+5 =... = v d1 = 0. Moreover, the Cartan test works!!!

60 A lengthy and technical computation shows (1 λ n) v λ = 1 2 (λ 1)(λ 2n 4)(λ 2n 5) v n+λ = 1 2 (n + λ 1)(n λ + 4)(n λ + 5) = 12n v 2n+1 v 2n+2 = 6n + 3 v 2n+3 = 2n + 2 v 2n+4 = v 2n+5 =... = v d1 = 0. Moreover, the Cartan test works!!! n = 1: D = = = 112.

61 A lengthy and technical computation shows (1 λ n) v λ = 1 2 (λ 1)(λ 2n 4)(λ 2n 5) v n+λ = 1 2 (n + λ 1)(n λ + 4)(n λ + 5) = 12n v 2n+1 v 2n+2 = 6n + 3 v 2n+3 = 2n + 2 v 2n+4 = v 2n+5 =... = v d1 = 0. Moreover, the Cartan test works!!! n = 1: D = = = 112. n = 2: D = = = 504.

62 The generality of the quaternionic contact structures Theorem The quaternionic contact structures in dimension 4n + 3 depend on 2n + 2 functions of 2n + 3 variables.

63 The generality of the quaternionic contact structures Theorem The quaternionic contact structures in dimension 4n + 3 depend on 2n + 2 functions of 2n + 3 variables. Moreover, given some arbitrary complex numbers S αβγδ, V αβγ, L αβ, M αβ, C α, H α, P, Q, R, A αβγδɛ, B αβγδ, C αβγδ, D αβγ, E αβγ, F αβγ, G αβ, X αβ, Y αβ, Z αβ, (N 1 ) α, (N 2 ) α, (N 3 ) α, (N 4 ) α, (N 5 ) α, U s, W s depending totally symmetrically on the indices 1 α, β, γ, δ 2n and satisfiyng (js ) αβγδ = S αβγδ, (jl ) αβ = L αβ, R = R, there exists a real analytic qc structure defined in a neighborhood Ω of 0 R 4n+3 such that for some point u P 1 with π o (π 1 (u)) = 0 the curvature functions and their derivatives are given by these complex numbers.

64 The generality of the quaternionic contact structures Theorem The quaternionic contact structures in dimension 4n + 3 depend on 2n + 2 functions of 2n + 3 variables. Moreover, given some arbitrary complex numbers S αβγδ, V αβγ, L αβ, M αβ, C α, H α, P, Q, R, A αβγδɛ, B αβγδ, C αβγδ, D αβγ, E αβγ, F αβγ, G αβ, X αβ, Y αβ, Z αβ, (N 1 ) α, (N 2 ) α, (N 3 ) α, (N 4 ) α, (N 5 ) α, U s, W s depending totally symmetrically on the indices 1 α, β, γ, δ 2n and satisfiyng (js ) αβγδ = S αβγδ, (jl ) αβ = L αβ, R = R, there exists a real analytic qc structure defined in a neighborhood Ω of 0 R 4n+3 such that for some point u P 1 with π o (π 1 (u)) = 0 the curvature functions and their derivatives are given by these complex numbers. It is interesting to compare this to an old result by Claude LeBrun (Duke Math. J., August 1991) where he finds a nontrivial