4-Dimensional Geometries  

In General
* Results: There are pairs of homeomorphic, non-diffeomorphic smooth 4-manifolds, in which one manifold admits an Einstein metric and the other does not, or they admit Einstein metrics with opposite signs of R; There are simply connected 4-manifolds for which the space of R > 0 Riemannian metrics is disconnected.
@ Differentiable structure and metric: Kotschick G&T(98)m.DG [homeomorphic manifolds and Einstein metrics].
@ Riemannian: Karlhede CQG(88) [classification]; Kapovich JDG(04) [conformally flat].
@ Other topics: Ruberman G&T(01)m.DG [R > 0 and diffeomorphisms].
> Online resources: see MathWorld page; Wikipedia page.

4D Lorentzian Geometry > s.a. lorentzian geometry; types of metrics.
@ General references: Grant & Vickers CQG(09)-a0809 [block diagonalisation].
@ Classification: Karlhede GRG(80); Milson & Pelavas CQG(08)-a0710 [type N].
> Physics applications: see general relativity; spacetime.

Special Cases > see differentiable manifolds [exotic 4-spheres]; euclidean geometry [4D euclidean geomery].

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