Higher-dimensional Osserman metrics with non-nilpotent Jacobi operators

Geometriae Dedicata (Impact Factor: 0.52). 07/2010; 156(1). DOI: 10.1007/s10711-011-9595-y
Source: arXiv


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 para-complex Osserman with non-trivial Jordan normal form as well; they neither satisfy the third Gray identity nor are they integrable.

Download full-text


Available from: Esteban Calviño-Louzao, Feb 08, 2014
  • Source
    • "Taking (M, ∇) to be non-flat gives rise to Osserman metrics on neutral signature manifolds with nonnilpotent Jacobi operators and with non-trivial Jordan normal form which admit natural para-Hermitian structures. They are semi para-complex space forms which neither satisfy the third Gray identity nor need they be integrable [11]. 1.3. "
    [Show abstract] [Hide abstract]
    ABSTRACT: A curvature model (V,A) is a real vector space V which is equipped with a "curvature operator" A(x,y)z that A has the same symmetries as an affine curvature operator; A(x,y)z=-A(y,x)z and A(x,y)z+A(y,z)x+A(z,x)y=0. Such a model is called projective affine Osserman if the spectrum of the Jacobi operator J(y):x->A(x,y)y, is projectively constant. There are topological conditions imposed on such a model by Adam's Theorem concerning vector fields on spheres. In this paper we construct projective affine Osserman curvature models when the dimension is odd, when the dimension is congruent to 2 mod 4, and when the dimension is congruent to 4 mod 8 for all the eigenvalue structure is allowed by Adam's Theorem.
    Preview · Article · Mar 2014 · Journal of Fixed Point Theory and Applications
  • Source
    • "It is possible to modify this construction to produce Osserman metrics with nonnilpotent Jacobi operators of neutral signature on T * M Calvino-Louzao et al [5]. "
    [Show abstract] [Hide abstract]
    ABSTRACT: By considering the projectivized spectrum of the Jacobi operator, we introduce the concept of projective Osserman manifold in both the affine and in the pseudo-Riemannian settings. If M is an affine projective Osserman manifold, then the deformed Riemannian extension metric on the cotangent bundle is both spacelike and timelike projective Osserman. Since any rank-1-symmetric space is affine projective Osserman, this provides additional information concerning the cotangent bundle of a rank-1 Riemannian symmetric space with the deformed Riemannian extension metric. We construct other examples of affine projective Osserman manifolds where the Ricci tensor is not symmetric and thus the connection in question is not the Levi-Civita connection of any metric. If the dimension is odd, we use methods of algebraic topology to show the Jacobi operator of an affine projective Osserman manifold has only one non-zero eigenvalue and that eigenvalue is real.
    Preview · Article · Jul 2013 · Classical and Quantum Gravity
  • Source
    • "Affine Osserman connections are of interest not only in affine geometry, but also in the study of pseudo-Riemannian Osserman metrics since they provide some nice examples without Riemannian analogue by means of the Riemannian extensions. Here it is worth to emphasize that some recent modifications of the usual Riemann extensions allowed some new applications [2] [3] [10] Let X = 3 i=1 α i ∂ i is a vector on a 3-dimensional affine manifold M, then the affine Jacobi operator is given by "
    [Show abstract] [Hide abstract]
    ABSTRACT: The aim of this note is to study the Osserman condition on two families affine connections. As applications, examples of affine Osserman connections which are Ricci flat but not flat are given.
    Full-text · Article · Feb 2012
Show more