10 2. QUATERNIONIC CONTACT STRUCTURES AND THE BIQUARD CONNECTION

1-dimensional, i.e. the [3]-component of any symmetric endomorphism Ψ on H is

proportional to the identity, Ψ3 =

tr(Ψ)

4

Id|H .

There exists a canonical connection compatible with a given quaternionic con-

tact structure. This connection was discovered by O. Biquard [Biq1] when the

dimension (4n + 3) 7 and by D. Duchemin [D] in the 7-dimensional case. The

next result due to O. Biquard is crucial in the quaternionic contact geometry.

Theorem 2.4. [Biq1] Let (M, g, Q) be a quaternionic contact manifold of di-

mension 4n+3 7 and a fixed metric g on H in the conformal class [g]. Then there

exists a unique connection ∇ with torsion T on M 4n+3 and a unique supplementary

subspace V to H in TM, such that:

i) ∇ preserves the decomposition H ⊕ V and the metric g;

ii) for X, Y ∈ H, one has T (X, Y ) = −[X, Y ]|V ;

iii) ∇ preserves the Sp(n)Sp(1)-structure on H, i.e., ∇g = 0 and ∇Q ⊂ Q;

iv) for ξ ∈ V , the endomorphism T (ξ, .)|H of H lies in (sp(n) ⊕ sp(1))⊥ ⊂ gl(4n);

v) the connection on V is induced by the natural identification ϕ of V with the

subspace sp(1) of the endomorphisms of H, i.e. ∇ϕ = 0.

In (iv) the inner product on End(H) is given by

(2.9) g(A, B) =

tr(B∗A)

=

4n

a=1

g(A(ea),B(ea)),

where A, B ∈ End(H), {e1, ..., e4n} is some g-orthonormal basis of H.

We shall call the above connection the Biquard connection. Biquard [Biq1] also

described the supplementary ”vertical” (sub-)space V explicitly, namely, locally V

is generated by vector fields {ξ1,ξ2,ξ3}, such that

(2.10)

ηs(ξk) = δsk, (ξs dηs)|H = 0,

(ξs dηk)|H = −(ξk dηs)|H , s, k ∈ {1, 2, 3}.

The vector fields ξ1,ξ2,ξ3 are called Reeb vector fields or fundamental vector fields.

If the dimension of M is seven, the conditions (2.10) do not always hold.

Duchemin shows in [D] that if we assume, in addition, the existence of Reeb vec-

tor fields as in (2.10), then Theorem 2.4 holds. Henceforth, by a QC structure in

dimension 7 we shall always mean a QC structure satisfying (2.10).

Notice that equations (2.10) are invariant under the natural SO(3) action.

Using the Reeb vector fields we extend g to a metric on M by requiring

(2.11) span{ξ1,ξ2,ξ3} = V ⊥ H and g(ξs,ξk) = δsk.

The extended metric does not depend on the action of SO(3) on V , but it changes

in an obvious manner if η is multiplied by a conformal factor. Clearly, the Biquard

connection preserves the extended metric on TM, ∇g = 0. We shall also extend

the quternionic structure by setting Is|V = 0.

Suppose the Reeb vector fields {ξ1,ξ2,ξ3} have been fixed. The restriction of

the torsion of the Biquard connection to the vertical space V satisfies [Biq1]

(2.12) T (ξi,ξj) = λξk − [ξi,ξj]|H ,

where λ is a smooth function on M, which will be determined in Corollary 3.14. We

recall that in the paper the indices follow the convention 1.4. Further properties of