Stana Nikcevic

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

Geometric realizations, curvature decompositions, and Weyl manifolds

Abstract We show that any Weyl curvature model can be geometrically realized by a Weyl manifold.

PSEUDO-RIEMANNIAN JACOBI–VIDEV MANIFOLDS

We exhibit several families of Jacobi-Videv pseudo-Riemannian manifolds which are not Einstein. We also exhibit Jacobi-Videv algebraic curvature tensors where the Ricci operator defines an almost complex structure.

Complete curvature homogeneous pseudo-Riemannian manifolds

Abstract. For k ≥ 2, we exhibit complete k-curvature homogeneous neutral signature pseudo-Riemannian manifolds which are not locally affine homogeneous (and hence not locally homogeneous). All the local scalarWeyl invariants of these manifolds vanish. These manifolds are Ricci flat, Osserman, and Ivanov-Petrova.

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.

CURVATURE STRUCTURE OF SELF-DUAL 4-MANIFOLDS

We show the existence of a modified Cliff(1,1)-structure compatible with an Osserman 0-model of signature (2,2). We then apply this algebraic result to certain classes of pseudo-Riemannian manifolds of signature (2,2). We obtain a new characterization of the Weyl curvature tensor of an (anti-)self-dual manifold and we prove some new results regarding (Jordan) Osserman 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.

AFFINE CURVATURE HOMOGENEOUS 3-DIMENSIONAL LORENTZ MANIFOLDS

We study a family of 3-dimensional Lorentz manifolds. Some members of the family are 0-curvature homogeneous, 1-affine curvature homogeneous, but not 1-curvature homogeneous. Some are 1-curvature homogeneous but not 2-curvature homogeneous. All are 0-modeled on indecomposible local symmetric spaces. Some of the members of the family are geodesically complete, others are not. All have vanishing scalar invariants.

Curvature homogeneous spacelike Jordan Osserman pseudo-Riemannian manifolds

Let s ≥ 2. We construct Ricci flat pseudo-Riemannian manifolds of signature (2s, s) which are not locally homogeneous but whose curvature tensors nevertheless exhibit a number of important symmetry properties. They are curvature homogeneous; their curvature tensor is modelled on that of a local symmetric space. They are spacelike Jordan Osserman with a Jacobi operator which is nilpotent of order 3; they are not timelike Jordan Osserman. They are k-spacelike higher-order Jordan Osserman for 2 ≤ k ≤ s; they are k-timelike higher-order Jordan Osserman for s + 2 ≤ k ≤ 2s, and they are not k-timelike higher-order Jordan Osserman for 2 ≤ k ≤ s + 1.

THE STRUCTURE OF THE SPACE OF AFFINE KÄHLER CURVATURE TENSORS AS A COMPLEX MODULE

In dimension m ≥ 4, results of Strichartz decompose the space 𝔄 of affine curvature tensors as a direct sum of 3 modules in the real setting and results of Bokan give a corresponding finer decomposition of 𝔄 in the Riemannian setting as the direct sum of 8 irreducible modules. In dimension m ≥ 8, results of Matzeu and Nikcevic decompose the space 𝔎 of affine Kahler curvature tensors as the direct sum of 12 irreducible modules in the Hermitian setting (i.e. given an auxiliary inner product which is invariant under the given almost complex structure). In this paper, we decompose 𝔎 as a direct sum of six irreducible modules in the complex setting in dimension m ≥ 8. Corresponding decompositions into fewer modules are given in dimension m = 4 and m = 6.