Related papers: Higher order Jordan Osserman Pseudo-Riemannian man…

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…

We construct a family of pseudo-Riemannian manifolds so that the skew-symmetric curvature operator, the Jacobi operator, and the Szabo operator have constant eigenvalues on their domains of definition. This provides new and non-trivial…

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…

We exhibit Osserman metrics with non-nilpotent Jacobi operators and with non-trivial Jordan normal form in neutral signature (n,n) for any n which is at least 3. These examples admit a natural almost para-Hermitian structure and are semi…

We exhibit 3 families of complete curvature homogeneous pseudo-Riemannian manifolds which are modeled on irreducible symmetric spaces and which are not locally homogeneous. All of the manifolds have nilpotent Jacobi operators; some of the…

We show that every paracomplex space form is locally isometric to a modified Riemannian extension and give necessary and sufficient conditions so that a modified Riemannian extension is Einstein. We exhibit Riemannian extension Osserman…

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…

Pseudo-Riemannian manifolds of balanced signature which are both spacelike and timelike Jordan Osserman nilpotent of order 2 and of order 3 have been constructed previously. In this short note, we shall construct pseudo-Riemannian manifolds…

We consider four-dimensional Riemannian manifolds with commuting higher order Jacobi operators defined on two-dimensional orthogonal subspaces (polygons) and on their orthogonal subspaces. More precisely, we discuss higher order Jacobi…

We study the geometry of pseudo-Riemannian manifolds which are Jacobi--Tsankov, i.e. J(x)J(y)=J(y)J(x) for all tangent vectors x and y. We also study manifolds which are 2-step Jacobi nilpotent, i.e. J(x)J(y)=0 for all tangent vectors x and…

In this paper, we study Jacobi operators associated to algebraic curvature maps (tensors) on lightlike submanifolds M. We investigate conditions for an induced Rie- mann curvature tensor to be an algebraic curvature tensor on M. We…

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.

Osserman manifolds are a generalization of locally two-point homogeneous spaces. We introduce $k$-root manifolds in which the reduced Jacobi operator has exactly $k$ eigenvalues. We investigate one-root and two-root manifolds as another…

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…

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…

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…

Let $R$ be an algebraic curvature tensor on a vector space of signature $(p,q)$ defining a spacelike Jordan Osserman Jacobi operator $\JJ_R$. We show that the eigenvalues of $\JJ_R$ are real and that $\JJ_R$ is diagonalizable if $p<q$.

One says that a smooth manifold M is a pseudo-Riemannian manifold of signature (p,q) if the tangent bundle TM is equipped with a smooth non-degenerate symmetric inner product g of signature (p,q). Similarly one says that M is an affine…

In this paper we define a Grassmann odd analogue of Jacobi structure on a supermanifold. The basic properties are explored. The construction of odd Jacobi manifolds is then used to reexamine the notion of a Jacobi algebroid. It is shown…

For a Riemannian manifold $M^n$ with the curvature tensor $R$, the Jacobi operator $R_X$ is defined by $R_XY = R(X,Y)X$. The manifold $M^n$ is called {\it pointwise Osserman} if, for every $p \in M^n$, the eigenvalues of the Jacobi operator…