Hitchin–Thorpe inequality

Proof

Let (M, g) be a four-dimensional smooth Riemannian manifold which is Einstein. Given any point p of M, there exists a g_p-orthonormal basis e₁, e₂, e₃, e₄ of the tangent space T_pM such that the curvature operator Rm_p, which is a symmetric linear map of ∧²T_pM into itself, has matrix

${\displaystyle \left(\begin{array}{cccccc}{\lambda }_1& 0& 0& {\mu }_1& 0& 0\\ 0& {\lambda }_2& 0& 0& {\mu }_2& 0\\ 0& 0& {\lambda }_3& 0& 0& {\mu }_3\\ {\mu }_1& 0& 0& {\lambda }_1& 0& 0\\ 0& {\mu }_2& 0& 0& {\lambda }_2& 0\\ 0& 0& {\mu }_3& 0& 0& {\lambda }_3\end{array}\right)}$

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

${\displaystyle \begin{array}{cc}\chi (M)& =\frac{1}{4{\pi }^2}{\int }_M({\lambda }_1^2+{\lambda }_2^2+{\lambda }_3^2+{\mu }_1^2+{\mu }_2^2+{\mu }_3^2)d{\mu }_g\\ \tau (M)& =\frac{1}{3{\pi }^2}{\int }_M({\lambda }_1{\mu }_1+{\lambda }_2{\mu }_2+{\lambda }_3{\mu }_3)d{\mu }_g.\end{array}}$

Equipped with these tools, the Hitchin-Thorpe inequality amounts to the elementary observation

${\displaystyle {\lambda }_1^2+{\lambda }_2^2+{\lambda }_3^2+{\mu }_1^2+{\mu }_2^2+{\mu }_3^2=\underset{\ge 0}{\underbrace{({\lambda }_1-{\mu }_1{)}^2+({\lambda }_2-{\mu }_2{)}^2+({\lambda }_3-{\mu }_3{)}^2}}+2({\lambda }_1{\mu }_1+{\lambda }_2{\mu }_2+{\lambda }_3{\mu }_3).}$

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. 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.