Complex Submanifolds of Quaternionic

  • Published 2006


1. Preliminaries on quaternionic Kähler manifolds We shall give a survey on (immersed) submanifolds having some interest to be considered into a quaternionic Kähler manifold. A quaternionic Kähler manifold will be denoted by (M̃, g̃, Q), where g̃ is the Riemannian metric and (g̃, Q) is the quaternionic Hermitian structure on the 4n-dimensional manifold M̃ ≡ M̃ ; the quaternionic structure Q, which is parallel with respect to the Levi-Civita connection ∇̃ = ∇e , is locally generated by an admissible almost hypercomplex basis H = (J1, J2, J3 = J1J2) and the following identities hold: ∇̃XJα = ωγ(X)Jβ − ωβ(X)Jγ , X ∈ TM where α, β, γ is a cyclic permutation of 1, 2, 3 and the ωα, α = 1, 2, 3, are local 1-forms (depending on the choice of an admissible basis (Jα)). See for example [23],[1] for a basic introduction. Let us recall also that (M̃, g̃) is an Einstein manifold and there is a decomposition of the curvature tensor

Cite this paper

@inproceedings{MANIFOLDS2006ComplexSO, title={Complex Submanifolds of Quaternionic}, author={K{\"AHLER MANIFOLDS and STEFANO MARCHIAFAVA}, year={2006} }