In differential geometry the Hitchin–Thorpe inequality is a famous relation which restricts the topology of 4-manifolds that carry an Einstein metric.

Contents 1 Statement of the Hitchin–Thorpe inequality 2 Idea of the proof 3 Failure of the converse 4 Footnotes 5 References

[edit] Statement of the Hitchin–Thorpe inequality

Let M be a compact, oriented, smooth four-dimensional manifold. If there exists a Riemannian metric on M which is an Einstein metric, then following inequality holds

where χ(M) is the Euler characteristic of M and τ(M) is the signature of M. This inequality was first stated by John Thorpe^[1] in a footnote to a 1969 paper focusing on manifolds of higher dimension. Nigel Hitchin then rediscovered the inequality, and gave a complete characterization ^[2] of the equality case in 1974; he found that if (M,g) is an Einstein manifold with then (M,g) must be a flat torus, a Calabi–Yau manifold, or a quotient thereof.

[edit] Idea of the proof

The main ingredients in the proof of the Hitchin–Thorpe inequality are the decomposition of the Riemann curvature tensor and the Generalized Gauss-Bonnet theorem.

[edit] Failure of the converse

A natural question to ask is whether the Hitchin–Thorpe inequality provides a sufficient condition for the existence of Einstein metrics. In 1995, Claude LeBrun and Andrea Sambusetti independently showed that the the answer is no: there exist infinitely many non-homeomorphic compact, smooth, oriented 4-manifolds M that carry no Einstein metrics but nevertheless satisfy

LeBrun's examples ^[3] are actually simply connected, and the relevant obstruction depends on the smooth structure of the manifold. By contrast, Sambusetti's obstruction ^[4] only applies to 4-manifolds with infinite fundamental group, but the volume-entropy estimate he uses to prove non-existence only depends on the homotopy type of the manifold.

[edit] Footnotes

1. ^ J. Thorpe, Some remarks on on the Gauss-Bonnet formula, J. Math. Mech. 18 (1969) pp. 779--786.
2. ^ N. Hitchin, On compact four-dimensional Einstein manifolds, J. Diff. Geom. 9 (1974) pp. 435--442.
3. ^ C. LeBrun, Four-manifolds without Einstein Metrics, Math. Res. Letters 3 (1996) pp. 133--147.
4. ^ A. Sambusetti, An obstruction to the existence of Einstein metrics on 4-manifolds, C.R. Acad. Sci. Paris 322 (1996) pp. 1213--1218.

[edit] References

• Besse, Arthur L. (1987). Einstein Manifolds. Classics in Mathematics. Berlin: Springer. ISBN 3-540-74120-8.