MM of HIV, HIV Population, and AIDS

Harmonic spinors of the Dirac operator of connection with torsion in dimension four

“…The key point is that the torsion 1-form transforms under conformal rescaling according to (3.2) (see [34], appendix 1). Application of this theorem leads to the following partial result of [23]. Proposition 3.1.…”

“…It turns out that a hyper-Hermitian manifold is HKT if and only if the Bismut connections of the three complex structures J α , α = 1, 2, 3, coincide [15,18,21]. In particular, the three Lee forms also coincide [17,23] and the common Lee form θ is called the Lee form of the HKT structure. If each J α , α = 1, 2, 3, is parallel with respect to the Levi-Civita connection then the structure is hyper-Kähler.…”

“…However, in the case of a QKT structure, i.e. when there is a quaternionic structure parallel with respect to ∇ and the torsion is (1, 2) + (2, 1)-form with respect to any compatible local almostcomplex structure (see, e.g., [28]), a natural 'conformal transformation' of the connection is defined in [23]. It is well known that every four-dimensional oriented Riemannian manifold carries a Hermitian quaternionic structure (see, e.g., [7]) and therefore every four-dimensional RT manifold is a QKT space in a natural way.…”

“…It is well known that every four-dimensional oriented Riemannian manifold carries a Hermitian quaternionic structure (see, e.g., [7]) and therefore every four-dimensional RT manifold is a QKT space in a natural way. Following [23] we define the conformally transformed connection ∇ by the equation…”

“…Following [23] we consider the associated Weyl structure, i.e. the unique torsion-free linear connection ∇ W preserving the conformal structure and determined by ∇ W g = * T ⊗ g, where * is the Hodge star operator.…”