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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send feedback and suggestions to bombelli at olemiss.edu – modified 1
jan
2016