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

0

+ bξ,

we summarize the description of the torsion due to O. Biquard in the following

Proposition.

Proposition 2.5. [Biq1] The torsion Tξ is completely trace-free,

(2.13) trTξ =

4n

a=1

g(Tξ(ea),ea) = 0, trTξ ◦ I =

4n

a=1

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:

(2.14) Tξi

0

Ii = −IiTξi

0

, i = 1, 2, 3.

In addition, we have

(2.15)

I2(Tξ2

0 )+−−

= I1(Tξ1

0 )−+−,

I3(Tξ3

0 )−+−

= I2(Tξ2

0 )−−+,

I1(Tξ1

0 )−−+

= I3(Tξ3

0 )+−−.

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

0.