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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

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

where \( \chi (M) \) is the Euler characteristic of M and \( {\displaystyle \tau (M)} \) is the signature of M {\displaystyle M} M. This inequality was first stated by John Thorpe in a footnote to a 1969 paper focusing on manifolds of higher dimension.[1] Nigel Hitchin then rediscovered the inequality, and gave a complete characterization of the equality case in 1974;[2] he found that if ( M , g ) {\displaystyle (M,g)} (M,g) is an Einstein manifold with \( {\displaystyle \chi (M)={\frac {3}{2}}|\tau (M)|,} \) 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

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

LeBrun's examples are actually simply connected, and the relevant obstruction depends on the smooth structure of the manifold.[3] 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.[4]
Footnotes

Thorpe, J. (1969). "Some remarks on the Gauss-Bonnet formula". J. Math. Mech. 18 (8): 779–786. JSTOR 24893137.
Hitchin, N. (1974). "Compact four-dimensional Einstein manifolds". J. Diff. Geom. 9 (3): 435–442. doi:10.4310/jdg/1214432419.
LeBrun, C. (1996). "Four-Manifolds without Einstein Metrics". Math. Res. Letters. 3 (2): 133–147. doi:10.4310/MRL.1996.v3.n2.a1.

Sambusetti, A. (1996). "An obstruction to the existence of Einstein metrics on 4-manifolds". C. R. Acad. Sci. Paris. 322 (12): 1213–1218. ISSN 0764-4442.

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

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