Isotropic Kähler structures on Engel 4-manifolds
The authors of this paper consider indefinite almost Hermitian manifolds \((M,g,J)\) whose almost complex structure \(J\) satisfies the following two conditions \(\nabla J\neq 0\) and \(\|\nabla J\|^2= 0\), \(\nabla\) being the Levi-Civita connection of the metric \(g\). When the second condition holds \(M\) is called an isotropic Kähler manifold. It is to remark that the second condition does not imply the Kähler condition \(\nabla J= 0\), i.e., the isotropic Kähler manifolds are not indefinite Kähler. Examples of non-Kähler isotropic Kähler manifolds are constructed in the case \(M= \mathbb{R}^4\) with an Engel structure (= a special absolutely non-integrable regular two-dimensional distribution). It is shown that \(M\) is endowed with two kinds of indefinite almost Hermitian structures \((g, J)\) and \((g,J')\) which are isotropic Kähler but neither complex nor symplectic. Isotropic Kähler almost Kähler manifolds are defined for which all linear invariants vanish.
scientific article; zbMATH DE number 1437068