STEFAN IVANOV, IVAN MINCHEV, and DIMITER VASSILEV 11
the Biquard connection are encoded in the properties of the torsion endomorphism
= T (ξ, .) : H H, ξ V.
Decomposing the endomorphism (sp(n) + sp(1))⊥ into its symmetric part Tξ0
and skew-symmetric part bξ,
=
0
+ bξ,
we summarize the description of the torsion due to O. Biquard in the following
Proposition.
Proposition 2.5. [Biq1] The torsion 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, =
0.
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