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 T 0 ξ 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) 0 i Ii = −IiTξ 0 i , i = 1, 2, 3. In addition, we have (2.15) I2(Tξ 0 2 )+−− = I1(Tξ 0 1 )−+−, I3(Tξ 0 3 )−+− = I2(Tξ 0 2 )−−+, I1(Tξ 0 1 )−−+ = I3(Tξ 0 3 )+−−. The skew-symmetric part can be represented in the following way (2.16) i = 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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