Eduardo García-Río

Osserman manifolds in semi-Riemannian geometry

The Osserman Conditions in Semi-Riemannian Geometry.- The Osserman Conjecture in Riemannian Geometry.- Lorentzian Osserman Manifolds.- Four-Dimensional Semi-Riemannian Osserman Manifolds with Metric Tensors of Signature (2,2).- Semi-Riemannian Osserman Manifolds.- Generalizations and Osserman-Related Conditions.

The Geometry of Walker Manifolds

* Basic Algebraic Notions* Basic Geometrical Notions* Walker Structures* Three-Dimensional Lorentzian Walker Manifolds* Four-Dimensional Walker Manifolds* The Spectral Geometry of the Curvature Tensor* Hermitian Geometry* Special Walker Manifolds

On a problem of Osserman in Lorentzian geometry

Abstract A problem of Osserman on the constancy of the eigenvalues of the Jacobi operator is studied in Lorentzian geometry. Attention is paid to the different cases of timelike, spacelike and null Osserman condition. One also shows a relation between the null Osserman condition and a previous one on infinitesimal null isotropy.

Four-dimensional Osserman metrics with nondiagonalizable Jacobi operators

A complete description of Osserman four-manifolds whose Jacobi operators have a nonzero double root of the minimal polynomial is given.

Three-dimensional Lorentz manifolds admitting a parallel null vector field

Curvature properties of three-dimensional Lorentz manifolds admitting a parallel degenerate line field are examined. A complete characterization of those manifolds being locally symmetric or locally conformally flat is obtained. The results of this study show nice families of examples of such properties within the Lorentzian setting.

Curvature properties of four-dimensional Walker metrics

A Walker n-manifold is a semi-Riemannian manifold, which admits a field of parallel null r-planes, with . In the present paper we study curvature properties of a Walker 4-manifold (M, g) which admits a field of parallel null 2-planes. The metric g is necessarily of neutral signature (+ + − −). Such a Walker 4-manifold is the lowest dimensional example not of Lorentz type. There are three functions of coordinates which define a Walker metric. Some recent work shows that a Walker 4-manifold of restricted type whose metric is characterized by two functions exhibits a large variety of symplectic structures, Hermitian structures, Kahler structures, etc. For such a restricted Walker 4-manifold, we shall study mainly curvature properties, e.g., conditions for a Walker metric to be Einstein, Osserman, or locally conformally flat, etc. One of our main results is the exact solutions to the Einstein equations for a restricted Walker 4-manifold.

Affine Osserman connections and their Riemann extensions

Abstract Osserman property is studied for affine torsion-free connections with special attention to the 2-dimensional case. As an application, examples of nonsymmetric and even not locally homogeneous Osserman pseudo-Riemannian metrics are constructed on the cotangent bundle of a manifold equipped with a torsionfree connection by looking at their Riemann extensions. Also, timelike and spacelike Osserman conditions are analyzed for general pseudo-Riemannian manifolds showing that they are equivalent.

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.

Isotropic Kähler structures on Engel 4-manifolds

Examples of isotropic Kahler manifolds (i.e., ∥∇J∥2=0) which are neither complex nor symplectic, and therefore not indefinite Kahler, are constructed.

Ricci solitons on Lorentzian manifolds with large isometry groups

We show that Lorentzian manifolds whose isometry group is of dimension at least 1 n(n � 1) + 1 admit different vector fields resulting in expanding, steady and shrinking Ricci solitons. Moreover, it is proved that those Ricci solitons are gradient (only) in the steady case. This provides examples of complete locally conformally flat and symmetric Lorentzian gradient Ricci solitons that are not rigid.