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In differential geometry the Hitchin–Thorpe inequality is a famous 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

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

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 in a footnote to a 1969 paper focusing on manifolds of higher dimension. Nigel Hitchin then rediscovered the inequality, and gave a complete characterization 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 are actually simply connected, and the relevant obstruction depends on the smooth structure of the manifold. By contrast, Sambusetti's obstruction 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.

References

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