E. Calviño-Louzao

The geometry of modified Riemannian extensions

We show that every paracomplex space form is locally isometric to a modified Riemannian extension and gives necessary and sufficient conditions for a modified Riemannian extension to be Einstein. We exhibit Riemannian extension Osserman manifolds of signature (3, 3), whose Jacobi operators have non-trivial Jordan normal form and which are not nilpotent. We present new four-dimensional results in Osserman geometry.

Riemann Extensions of Torsion-Free Connections with Degenerate Ricci Tensor

Correspondence between torsion-free connections with nilpotent skew-symmetric curvature operator and IP Riemann extensions is shown. Some consequences are derived in the study of four-dimensional IP metrics and locally homogeneous affine surfaces. Received by the editors April 29, 2008. Published electronically July 6, 2010. Supported by projects MTM2006-01432 and PGIDIT06PXIB207054PR (Spain). AMS subject classification: 53B30, 53C50.

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.

A note on compact Cotton solitons

Compact Riemannian Cotton solitons are shown to be locally conformally flat. Moreover, non-trivial Cotton solitons are constructed both in the compact Lorentzian and in the complete non-compact Riemannian settings.

Four-dimensional Osserman–Ivanov–Petrova metrics of neutral signature

Algebraic curvature tensors which are Osserman–IP in the (− − + +)-signature setting are completely determined. As a consequence, it is shown that a four-dimensional pointwise Osserman–IP manifold is a space of constant sectional curvature or, otherwise, at each point the Jacobi operators either vanish or they are two-step nilpotent.

Bach-flat isotropic gradient Ricci solitons

We construct examples of Bach-flat gradient Ricci solitons which are neither half conformally flat nor conformally Einstein.

Geometric properties of generalized symmetric spaces

It is shown that four-dimensional generalized symmetric spaces can be naturally equipped with some additional structures defined by means of their curvature operators. As an application, those structures are used to characterize generalized symmetric spaces.

Four‐dimensional Osserman metrics revisited

Four‐dimensional Osserman metrics are reviewed by focusing on their connection with Einstein self‐dual structures. Special attention is paid to the nondiagonalizability of the self‐dual Weyl curvature operator.

Local Structure of Self-Dual Gradient Yamabe Solitons

We analyze the underlying structure of a pseudo-Riemannian manifold admitting a gradient Yamabe soliton. Special attention is paid to neutral signature, where a description of self-dual gradient Yamabe solitons is obtained.