Hitchin–Thorpe inequality

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

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

$\chi(M) \geq \frac{3}{2}|\tau(M)|,$

where $\chi(M)$ is the Euler characteristic of $M$ and $\tau(M)$ 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 $\chi(M) = \frac{3}{2}|\tau(M)|,$ then $(M,g)$ must be a flat torus, a Calabi–Yau manifold, or a quotient thereof.

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.

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 answer is no: there exist infinitely many non-homeomorphic compact, smooth, oriented 4-manifolds M that carry no Einstein metrics but nevertheless satisfy

$\chi(M) > \frac{3}{2}|\tau(M)|.$

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.

Footnotes

1. J. Thorpe, Some remarks 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.

References

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