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

Geometriae Dedicata (Impact Factor: 0.47). 07/2010; DOI: 10.1007/s10711-011-9595-y

Source: arXiv

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Esteban Calviño-Louzao, Feb 08, 2014 Available from: Data provided are for informational purposes only. Although carefully collected, accuracy cannot be guaranteed. The impact factor represents a rough estimation of the journal's impact factor and does not reflect the actual current impact factor. Publisher conditions are provided by RoMEO. Differing provisions from the publisher's actual policy or licence agreement may be applicable.

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**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.Classical and Quantum Gravity 07/2013; 30(15):155015. DOI:10.1088/0264-9381/30/15/155015 · 3.10 Impact Factor - [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.Journal of Fixed Point Theory and Applications 03/2014; 16(1-2). DOI:10.1007/s11784-014-0203-2 · 0.57 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The present paper deals with the existence of new class of affine Osserman connections which are Ricci flat but not flat.