New examples of Osserman metrics with nondiagonalizable Jacobi operators

Explicit examples of Osserman 4-manifolds with exactly two distinct eigenvalues of the Jacobi operators, α and β=4α≠0, are given. The former has multiplicity two and is a double root of the minimal polynomial of the Jacobi operators.     

Nonsymmetric Osserman pseudo-Riemannian manifolds

Examples of Osserman pseudo-Riemannian manifolds with metric of any signature , , which are not locally symmetric are exhibited.

     

One-harmonic invariant vector fields on three-dimensional Lie groups

We determine all left-invariant vector fields on three-dimensional Lie groups which define harmonic sections of the corresponding tangent bundles, equipped with the complete lift metric.     

Homogeneous Ricci almost solitons

It is shown that a locally homogeneous proper Ricci almost soliton is either of constant sectional curvature or locally isometric to a product R×N(c), where N(c) is a space of constant curvature.     

Local rigidity and nonrigidity of symplectic pairs

It is shown that indefinite strictly almost K?hler and opposite K?hler structures (J, J′) on a four-dimensional manifold with J-invariant Ricci operator are rigid, thus extending a previous result of Apostolov, Armstrong and Drăghici from the positive definite case to the indefinite one. In contrast to this, examples of nonhomogeneous four-dimensional manifolds which admit strictly almost paraK?hler and opposite paraK?hler structures (\mathfrakJ,\mathfrakJ^￠){(\mathfrak{J},\mathfrak{J}^{\prime})} with \mathfrakJ{\mathfrak{J}} -invariant Ricci operator are shown.     

Higher-dimensional Osserman metrics with non-nilpotent Jacobi operators

We exhibit Osserman metrics with non-nilpotent Jacobi operators and with non-trivial Jordan normal form in neutral signature (n, n) for any n ≥ 3. These examples admit a natural almost para-Hermitian structure and are semi para-complex Osserman with non-trivial Jordan normal form as well; they neither satisfy the third Gray identity nor are they integrable.     

Locally conformally flat Kähler and para-Kähler manifolds

We complete the classification of locally conformally flat Kähler and para-Kähler manifolds, describing all possible non-flat curvature models for Kähler and para-Kähler surfaces.