Statement of the Hitchin–Thorpe inequality
where is the Euler characteristic of and is the signature of . 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 is an Einstein manifold with then must be a flat torus, a Calabi–Yau manifold, or a quotient thereof.
Idea of the proof
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
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.
- J. Thorpe, Some remarks on the Gauss-Bonnet formula, J. Math. Mech. 18 (1969) pp. 779--786.
- N. Hitchin, On compact four-dimensional Einstein manifolds, J. Diff. Geom. 9 (1974) pp. 435--442.
- C. LeBrun, Four-manifolds without Einstein Metrics, Math. Res. Letters 3 (1996) pp. 133--147.
- A. Sambusetti, An obstruction to the existence of Einstein metrics on 4-manifolds, C.R. Acad. Sci. Paris 322 (1996) pp. 1213--1218.