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
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