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.

Read more about this topic:  Einstein Manifold

Famous quotes containing the word examples:

    “There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
    —Bernard Mandeville (1670–1733)

    “It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
    —G.C. (Georg Christoph)