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.

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

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