8 2. QUATERNIONIC CONTACT STRUCTURES AND THE BIQUARD CONNECTION
¯
I s, s = 1, 2, 3, then ¯ η = μ Ψ η for some Ψ SO(3) and a positive function μ Typ-
ical examples of manifolds with QC-structures are totally umbilical hypersurfaces
in quaternionic ahler or hyperk¨ ahler manifold, see Proposition 6.12 for the latter.
It is instructive to consider the case when there is a globally defined one-form
η. The obstruction to the global existence of η is encoded in the first Pontrjagin
class [AK]. Besides clarifying the notion of a QC-manifold, most of the time, for
example when considering the Yamabe equation, we shall work with a QC-structure
for which we have a fixed globally defined contact form. In this case, if we rotate the
R3-valued contact form and the almost complex structures by the same rotation we
obtain again a contact form, almost complex structures and a metric (the latter is
unchanged) satisfying the above conditions. On the other hand, it is important to
observe that given a contact form the almost complex structures and the horizontal
metric are unique if they exist. Finally, if we are given the horizontal bundle and
a metric on it, there exists at most one sphere of associated contact forms with a
corresponding sphere Q of almost complex structures [Biq1]. These observations
are summed-up in the next Lemma.
Lemma 2.2. [Biq1]
a) If (η, Is,g) and (η, Is,g ) are two QC structures on M, then Is = Is, s = 1, 2, 3
and g = g .
b) If (η, g) and , g) are two QC structures on M with Ker(η) = Ker(η ) = H
then Q = Q and η = Ψ η for some matrix Ψ SO(3) with smooth functions
as entries.
Proof. a) Let us fix a basis {e1, ..., e4n} of H . Suppose the tensors g, dη1|H ,
dη2|H , dη3|H , I1,I2,I3 ( tensors on H) are given in local coordinates, respectively,
by the matrices G, N1,N2,N3,J1,J2,J3 GL(4n). From (2.1) it follows
(2.2) GJs =
1
2
Ns, s = 1, 2, 3.
Let (i, j, k) be any cyclic permutation of (1, 2, 3). Using (2.2) we compute that
Jk = JiJj =
−Ji−1G−1GJj
=
−(GJi)−1(GJj)
=
−Ni−1Nj,
(2.3)
which accomplishes the proof.
b) The condition Ker(η) = Ker(η ) = H implies that
(2.4) ηk =
3
l=1
Ψkl ηl, k = 1, 2, 3
for some matrix Ψ GL(3) with smooth functions Ψij as entries. Applying the
exterior derivative in (2.4) we find
(2.5) dηk = dΨkl ηl + Ψkl dηl, k = 1, 2, 3.
Let the H tensors Ik and Ik be defined as usual with (2.1) using respectively η and
η . Restricting the equation (2.5) to H and using the metric tensor g on H we have
(2.6) g(IkX, Y ) = Ψklg(IlX, Y ), X, Y H
or the equivalent equations Ik = ΨklIl on H. It is easy to see that this is possible
if and only if Ψ SO(3).
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