Homothety Curvature Homogeneity and Homothety Homogeneity

We examine the difference between several notions of curvature homogeneity and show that the notions introduced by Kowalski and Vanzurova are genuine generalizations of the ordinary notion of k-curvature homogeneity. The homothety group plays an essential role in the analysis. We give a complete classification of homothety homogeneous manifolds which are not homogeneous and which are not VSI and show that such manifolds are cohomogeneity one. We also give a complete description of the local geometry if the homothety character defines a split extension.

(para)-Kahler Weyl structures

We work in both the complex and in the para-complex categories and examine (para)-Kahler Weyl structures in both the geometric and in the algebraic settings. The higher dimensional setting is quite restrictive. We show that any (para)-Kaehler Weyl algebraic curvature tensor is in fact Riemannian in dimension at least 6; this yields as a geometric consequence that any (para)-Kaehler Weyl geometric structure is trivial if the dimension is at least 6. By contrast, the 4 dimensional setting is, as always, rather special as it turns out that there are (para)-Kaehler Weyl algebraic curvature tensors which are not Riemannian in dimension 4. Since every (para)-Kaehler Weyl algebraic curvature tensor is geometrically realizable and since every 4 dimensional Hermitian manifold admits a unique (para)-Kaehler Weyl structure, there are also non-trivial 4 dimensional Hermitian (para)-Kaehler Weyl manifolds.

Kahler-Weyl manifolds of dimension 4

We determine the space of algebraic pseudo-Hermitian Kahler-Weyl curvature tensors and the space of para-Hermitian Kahler-Weyl curvature tensors in dimension 4 and show that every algebraic possibility is geometrically realizable. We establish the Gray identity for pseudo-Hermitian Weyl manifolds and for para-Hermitian Weyl manifolds in arbitrary dimension.

The structure of the space of affine Kaehler curvature tensors as a complex module

We use results of Matzeu and Nikcevic to decompose the space of affine Kaehler curvature tensors as a direct sum of irreducible modules in the complex setting

Curvature properties of Weyl geometries

We examine relations between geometry and the associated curvature decompositions in Weyl geometry.

Geometric theory of equiaffine curvature tensors

We present an algebraic investigation of generalized and equiaffine curvature tensors in a given pseudo-Euclidean vector space and study different orthogonal, irreducible decompositions in analogy to the known decomposition of algebraic curvature tensors. We apply the decomposition results to characterize geometric properties of Codazzi structures and relative hypersurfaces; particular emphasis is on projectively flat structures.

Geometric Realizations of para-Hermitian curvature models

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.

Riemannian geometric realizations for Ricci tensors of generalized algebraic curvature operators

We examine questions of geometric realizability for algebraic structures which arise naturally in affine and Riemannian geometry. Suppose given an algebraic curvature operator R at a point P of a manifold M and suppose given a real analytic (resp. C-k for finite k at least 2) pseudo-Riemannian metric on M defined near P. We construct a torsion free real analytic (resp. C-k) connection D which is defined near P on the tangent bundle of M whose curvature operator is the given operator R at P and so that D has constant scalar curvature. We show that if R is Ricci symmetric, then D can be chosen to be Ricci symmetric; if R has trace free Ricci tensor, then D can be chosen to have trace free Ricci tensor; if R is Ricci alternating, then D can be chosen to be Ricci alternating.

Geometric realizations of generalized algebraic curvature operators

We study the 8 natural GL equivariant geometric realization questions for the space of generalized algebraic curvature tensors. All but one of them is solvable; a non-zero projectively flat Ricci antisymmetric generalized algebraic curvature is not geometrically realizable by a projectively flat Ricci antisymmetric torsion free connection.

The classification of simple Jacobi--Ricci commuting algebraic curvature tensors

We classify algebraic curvature tensors such that the Ricci operator is simple (i.e. the Ricci operator is complex diagonalizable and either the complex spectrum consists of a single real eigenvalue or the complex spectrum consists of a pair of eigenvalues which are complex conjugates of each other) and which are Jacobi--Ricci commuting (i.e. the Ricci operator commutes with the Jacobi operator of any vector).