Four-dimensional Osserman metrics with nondiagonalizable Jacobi operators

Abstract

A complete description of Osserman four-manifolds whose Jacobi operators have a nonzero double root of the minimal polynomial is given.2000 M. S. C.: 53C50, 53B30

Four-dimensional neutral signature self-dual gradient Ricci solitons

“…Moreover, a geometrical interpretation of the Einstein self-dual condition in terms of the spectrum of the Jacobi operators was investigated as the first non-trivial case of Osserman manifolds (cf. [3,12,14]). We emphasize here that the geometry of (anti)-self-dual conformal structures is much richer in neutral signature (2,2) than the corresponding Riemannian one (see, for example [17] and the references therein).…”

“…This provides an infinite family of gradient Ricci solitons associated to each affine gradient Ricci soliton. Also note that, while the Riemannian extension (T * Σ, g D ) is locally conformally flat if and only if (Σ, D) is projectively flat, (T * Σ, g D,Φ ) is always self-dual but not locally conformally flat even if (Σ, D) is projectively flat [8,12]. Moreover, all deformed Riemannian extensions in Theorem 1-(2) have zero scalar curvature and, hence, their self-dual Weyl curvature operator W + is always nilpotent [12].…”

“…Also note that, while the Riemannian extension (T * Σ, g D ) is locally conformally flat if and only if (Σ, D) is projectively flat, (T * Σ, g D,Φ ) is always self-dual but not locally conformally flat even if (Σ, D) is projectively flat [8,12]. Moreover, all deformed Riemannian extensions in Theorem 1-(2) have zero scalar curvature and, hence, their self-dual Weyl curvature operator W + is always nilpotent [12]. An application of Theorem 1 and results in [1] provide the following description of four-dimensional locally conformally flat gradient Ricci solitons with no prescribed signature.…”

Compact Osserman Manifolds with Neutral Metric

“…On the other hand, there are plenty of examples of Osserman metrics in signature (− − ++) (cf. [2,5,8,13,14,16]). The Jordan normal form plays a crucial role in the higher signature setting-a self-adjoint linear transformation need not be determined by its eigenvalues if the metric in question is indefinite.…”

Four-dimensional Osserman metrics of neutral signature

“…However the situation is much more complicated in neutral signature (2, 2); there exist many examples of nonsymmetric Osserman pseudo-Riemannian manifolds of neutral signature -see [Díaz-Ramos et al 2006b] and [García-Río et al 1998]. Despite the results of [Alekseevsky et al 1999;Blažić et al 2001;Díaz-Ramos et al 2006a;García-Río and Vázquez-Lorenzo 2001], it is still an open problem to completely describe 4-dimensional Osserman metrics of neutral signature.…”