Metrics Project

This page will eventually be a repository of all metrics on 4-manifolds. In time there will be dozens of explicit metrics and explanations of numerous techniques. Currently things are sparse, but with an upload or two per week, the project should be done in no time.

Primer: Metrics on 4-manifolds

Metrics arranged by type

• • Conformal Metrics and Metrics expressed in coordinates
    • Primer: Conformal changes of metric
    • The round sphere
    • Hyperbolic space
    • Half-conformally flat ALE spaces
    • Gravitationally inspired metrics
  • 4-metrics from frames on the 3-Sphere
    • Primer: Squashed 3-Spheres
    • The Taub-NUT family of metrics
    • Complete metrics on bundles over 2-spheres
    • Compact metrics
  • Homogeneity-one metrics

Remark. Not included are metrics that are known to exist but cannot be written down in principle; typically these are metrics whose discovery involves the implicit function theorem and/or a continuity argument. Examples of what is not included are Calabi-Yau metrics, metrics from the Kummer construction such as the Biquard-Minerbe metrics, the Gross-Wilson metrics on K3, and the Chen-Lebrun-Weber metric on CP²#2CP².

Remark. Sources cited inline.