Related papers: Algebraic curvature tensors whose skew-symmetric c…

We classify the connected pseudo-Riemannian manifolds of signature $(p,q)$ with $q\ge5$ so that at each point of $M$ the skew-symmetric curvature operator has constant rank 2 and constant Jordan normal form on the set of spacelike 2 planes…

We construct almost complex algebraic curvature tensors for pseudo Hermitian inner products whose skew-symmetric curvature operator has constant Jordan normal form on the set of non-degenerate complex lines.

Algebraic curvature tensors possess generators which can be formed from symmetric or alternating tensors S, A or tensors \theta with an irreducible (2,1)-symmetry. In differential geometry examples of curvature formulas are known which…

We construct new examples of algebraic curvature tensors so that the Jordan normal form of the higher order Jacobi operator is constant on the Grassmannian of subspaces of type $(r,s)$ in a vector space of signature $(p,q)$. We then use…

The paper is constructed in two parts.In the first part we introduce the concept of the algebra of Q-meromorphic functions on the quantum plane.The A (q)-algebra of Q-analytic functions considered in[6]is seen as a proper subalgebra. In the…

Let E be a natural operator associated to the curvature tensor of a pseudo-Riemannian manifold. This survey article studies when the spectrum, or more generally the real Jordan normal form, of E is constant on the natural domain of…

Let M be a pseudo-Riemannian manifold with a pseudo-Hermitian complex structure $J$. We give necessary and sufficient conditions that the curvature operator $R(\pi)$ is complex linear when $\pi$ is a $J$ invariant real 2 plane. Under this…

Together with spaces of constant sectional curvature and products of a real line with a manifold of constant curvature, the socalled Egorov spaces and $\varepsilon$-spaces exhaust the class of $n$-dimensional Lorentzian manifolds admitting…

An algebraic curvature tensor A is said to be Jacobi-Tsankov if J(x)J(y)=J(y)J(x) for all x,y. This implies J(x)J(x)=0 for all x; necessarily A=0 in the Riemannian setting. Furthermore, this implies J(x)J(y)=0 for all x,y if the dimension…

We consider a 4-dimensional Riemannian manifold M endowed with a right skew-circulant tensor structure S, which is an isometry with respect to the metric g and the fourth power of S is minus identity. We determine a class of manifolds (M,…

We present some examples of curvature homogeneous pseudo-Riemannian manifolds which are k-spacelike Jordan Stanilov; their higher order curvature operator has constant Jordan normal form on the Grassmannian of unoriented k-dimensional…

We show that any $k$ Osserman Lorentzian algebraic curvature tensor has constant sectional curvature and give an elementary proof that any local 2 point homogeneous Lorentzian manifold has constant sectional curvature. We also show that a…

The projective curvature tensor $P$ is invariant under a geodesic preserving transformation on a semi-Riemannian manifold. It is well known that $P$ is not a generalized curvature tensor and hence it possesses different geometric properties…

Let $J(\pi)$ be the higher order Jacobi operator. We study algebraic curvature tensors where $J(\pi)J(\pi^{\perp})=J(\pi^{\perp})J(\pi)$. In the Riemannian setting, we give a complete characterization of such tensors; in the…

A pseudo-Riemannian manifold is said to be spacelike Jordan IP if the Jordan normal form of the skew-symmetric curvature operator depends upon the point of the manifold, but not upon the particular spacelike 2-plane in the tangent bundle at…

We classify the algebraic curvature tensors which are both Osserman and complex Osserman in all but a finite number of exceptional dimensions.Information concerning the possible eigenvalue structures, which is provided by methods of…

We show that if $\nabla R$ is a Jordan Szabo algebraic covariant derivative curvature tensor on a vector space of signature (p,q), where q is odd and p is less than q or if q is congruent to 2 mod 4 and if p is less than q-1, then $\nabla…

We use curvature decompositions to construct generating sets for the space of algebraic curvature tensors and for the space of tensors with the same symmetries as those of a torsion free, Ricci symmetric connection; the latter naturally…

Let s be at least 2. We construct Ricci flat pseudo-Riemannian manifolds of signature (2s,s) which are not locally homogeneous but whose curvature tensors never the less exhibit a number of important symmetry properties. They are curvature…

For a positive definite fundamental tensor all known examples of Osserman algebraic curvature tensors have a typical structure. They can be produced from a metric tensor and a finite set of skew-symmetric matrices which fulfil Clifford…