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 closed, oriented, four-dimensional smooth manifold. If there exists a Riemannian metric on M which is an Einstein metric, then
χ ( M ) ≥ 3 2 | τ ( M ) | , {\displaystyle \chi (M)\geq {\frac {3}{2}}|\tau (M)|,}
where χ(M) is the Euler characteristic of M and τ(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 for which equality in the Hitchin-Thorpe inequality is obtained, then the Ricci curvature of g is zero; if the sectional curvature is not identically equal to zero, then (M, g) is a Calabi–Yau manifold whose universal cover is a K3 surface. Already in 1961, Marcel Berger showed that the Euler characteristic is always non-negative.
Proof Let (M, g) be a four-dimensional smooth Riemannian manifold which is Einstein. Given any point p of M, there exists a gp-orthonormal basis e1, e2, e3, e4 of the tangent space TpM such that the curvature operator Rmp, which is a symmetric linear map of ∧2TpM into itself, has matrix
( λ 1 0 0 μ 1 0 0 0 λ 2 0 0 μ 2 0 0 0 λ 3 0 0 μ 3 μ 1 0 0 λ 1 0 0 0 μ 2 0 0 λ 2 0 0 0 μ 3 0 0 λ 3 ) {\displaystyle {\begin{pmatrix}\lambda _{1}&0&0&\mu _{1}&0&0\\0&\lambda _{2}&0&0&\mu _{2}&0\\0&0&\lambda _{3}&0&0&\mu _{3}\\\mu _{1}&0&0&\lambda _{1}&0&0\\0&\mu _{2}&0&0&\lambda _{2}&0\\0&0&\mu _{3}&0&0&\lambda _{3}\end{pmatrix}}}
relative to the basis e1 ∧ e2, e1 ∧ e3, e1 ∧ e4, e3 ∧ e4, e4 ∧ e2, e2 ∧ e3. One has that μ1 + μ2 + μ3 is zero and that λ1 + λ2 + λ3 is one-fourth of the scalar curvature of g at p. Furthermore, under the conditions λ1 ≤ λ2 ≤ λ3 and μ1 ≤ μ2 ≤ μ3, each of these six functions is uniquely determined and defines a continuous real-valued function on M. According to Chern-Weil theory, if M is oriented then the Euler characteristic and signature of M can be computed by
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