STEFAN IVANOV, IVAN MINCHEV, and DIMITER VASSILEV 11
the Biquard connection are encoded in the properties of the torsion endomorphism
Tξ = T (ξ, .) : H → H, ξ ∈ V.
Decomposing the endomorphism Tξ ∈ (sp(n) + sp(1))⊥ into its symmetric part Tξ0
and skew-symmetric part bξ,
Tξ = Tξ
we summarize the description of the torsion due to O. Biquard in the following
Proposition 2.5. [Biq1] The torsion Tξ is completely trace-free,
(2.13) trTξ =
g(Tξ(ea),ea) = 0, trTξ ◦ I =
g(Tξ(ea),Iea) = 0, I ∈ Q,
where e1 . . . e4n is an orthonormal basis of H. The symmetric part of the torsion
has the properties:
Ii = −IiTξi
, i = 1, 2, 3.
In addition, we have
The skew-symmetric part can be represented in the following way
(2.16) bξi = Iiu, i = 1, 2, 3,
where u is a traceless symmetric (1,1)-tensor on H which commutes with I1,I2,I3.
If n = 1 then the tensor u vanishes identically, u = 0 and the torsion is a
symmetric tensor, Tξ = Tξ