Article

# Higher-dimensional Osserman metrics with non-nilpotent Jacobi operators

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

ABSTRACT

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.

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Available from: Esteban Calviño-Louzao, Feb 08, 2014
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• "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. "
##### Article: Projective affine Osserman curvature models
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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.
Journal of Fixed Point Theory and Applications 03/2014; 16(1-2). DOI:10.1007/s11784-014-0203-2 · 0.55 Impact Factor
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• "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]. "
##### Article: Affine projective Osserman structures
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Classical and Quantum Gravity 07/2013; 30(15):155015. DOI:10.1088/0264-9381/30/15/155015 · 3.17 Impact Factor
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• "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 "
##### Article: Two families of affine Osserman connections on3-dimensional manifolds
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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.