Einstein Manifold - Examples

Examples

Simple examples of Einstein manifolds include:

  • Any manifold with constant sectional curvature is an Einstein manifold—in particular:
    • Euclidean space, which is flat, is a simple example of Ricci-flat, hence Einstein metric.
    • The n-sphere, Sn, with the round metric is Einstein with k = n − 1.
    • Hyperbolic space with the canonical metric is Einstein with negative k.
  • Complex projective space, CPn, with the Fubini-Study metric.
  • Calabi Yau manifolds admit a unique Einstein metric that is also Kähler.

A necessary condition for closed, oriented, 4-manifolds to be Einstein is satisfying the Hitchin–Thorpe inequality.

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