ar X iv : m at h / 06 12 80 4 v 3 [ m at h . D G ] 1 7 A ug 2 00 8 A Spinor Approach to Walker Geometry

ar X iv : m at h / 06 12 80 4 v 4 [ m at h . D G ] 7 A pr 2 00 9 A Spinor Approach to Walker Geometry

A four-dimensional Walker geometry is a four-dimensional manifold M with a neutral metric g and a parallel distribution of totally null two-planes. This distribution has a natural characterization as a projective spinor field subject to a certain constraint. Spinors therefore provide a natural tool for studying Walker geometry, which we exploit to draw together several themes in recent explicit...

ar X iv : 0 80 3 . 44 39 v 2 [ m at h . D S ] 4 A ug 2 00 8 PERIODIC UNIQUE BETA - EXPANSIONS : THE SHARKOVSKIĬ ORDERING

Let β ∈ (1, 2). Each x ∈ [0, 1 β−1 ] can be represented in the form

ar X iv : 0 80 4 . 21 32 v 1 [ m at h . D G ] 1 4 A pr 2 00 8 GEOMETRY OF NEUTRAL METRICS IN DIMENSION FOUR ∗

The purpose of this article is to review some recent results on the geometry of neutral signature metrics in dimension four and their twistor spaces. The following topics are considered: Neutral Kähler and hyperkähler surfaces, Walker metrics, Neutral anti-self-dual 4-manifolds and projective structures, Twistor spaces of neutral metrics.

ar X iv : h ep - t h / 06 12 12 8 v 1 1 3 D ec 2 00 6 Riemannian Geometry of Noncommutative Surfaces

ar X iv : m at h / 06 08 16 3 v 1 [ m at h . L O ] 7 A ug 2 00 6 IND - AND PRO - DEFINABLE SETS

We describe the indand procategories of the category of definable sets, in some first order theory, in terms of points in a sufficiently saturated model.