Miguel Brozos-Vázquez

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

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.

Conformally Osserman four-dimensional manifolds whose conformal Jacobi operators have complex eigenvalues

Conformal Osserman four-dimensional manifolds are studied with special attention to the construction of new examples showing that the algebraic structure of any such curvature tensor can be realized at the differentiable level. As a consequence one gets examples of anti-self-dual manifolds whose anti-self-dual curvature operator has complex eigenvalues.

Some remarks on locally conformally flat static space–times

Necessary and sufficient conditions for a static space–time to be locally conformally flat are obtained, showing some significant restrictions on the possible warping functions of the space–times. This occurs in opposition to cosmological models, where Robertson–Walker space–times are locally conformally flat for any warping function.

Geometric Realizations of Para-Hermitian Curvature Models

Abstract.We show that a para-Hermitian algebraic curvature model satisfies the para-Gray identity if and only if it is geometrically realizable by a para-Hermitian manifold. This requires extending the Tricerri–Vanhecke curvature decomposition to the para-Hermitian setting. Additionally, the geometric realization can be chosen to have constant scalar curvature and constant *-scalar curvature.

Relating the curvature tensor and the complex Jacobi operator of an almost Hermitian manifold

Let J be a unitary almost complex structure on a Riemannian manifold (M, g). If x is a unit tangent vector, let π := Span{x, Jx} be the associated complex line in the tangent bundle of M . The complex Jacobi operator and the complex curvature operators are defined, respectively, by J (π) := J (x) + J (Jx) and R(π) := R(x, Jx). We show that if (M, g) is Hermitian or if (M,g) is nearly Kahler, then either the complex Jacobi operator or the complex curvature operator completely determine the full curvature operator; this generalizes a well known result in the real setting to the complex setting. We also show this result fails for general almost Hermitian manifolds.

Stanilov-Tsankov-Videv Theory

We survey some recent results concerning Stanilov-Tsankov-Videv theory, con- formal Osserman geometry, and Walker geometry which relate algebraic properties of the curvature operator to the underlying geometry of the manifold.

Pseudo-Riemannian manifolds with commuting "Jacobi operators"

We study the geometry of pseudo-Riemannian manifolds which are Jacobi-Tsankov, i.e. ℊ(x)ℊ(y)=ℊ(y)ℊ(x) for allx, y. We also study manifolds which are 2-step Jacobi nilpotent, i.e. ℊ(x)ℊ(y)=0 for allx, y.

Examples of signature (2, 2) manifolds with commuting curvature operators

We exhibit Walker manifolds of signature (2, 2) with various commutativity properties for the Ricci operator, the skew-symmetric curvature operator and the Jacobi operator. If the Walker metric is a Riemannian extension of an underlying affine structure , these properties are related to the Ricci tensor of .

Locally conformally flat multidimensional cosmological models and generalized Friedmann–Robertson–Walker spacetimes

Multidimensional cosmological solutions which are locally conformally flat are described, thus leading to generalizations of the Friedmann–Robertson–Walker cosmology.