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.

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

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 $g_{p}$-orthonormal basis $e_{1}, e_{2}, e_{3}, e_{4}$ of the tangent space $T_{p}M$ such that the curvature operator $Rm_{p}$, which is a symmetric linear map of $∧^{2}T_{p}M$ into itself, has matrix
 * $$\begin{pmatrix}\lambda_1&0&0&\mu_1&0&0\\ 0&\lambda_2&0&0&\mu_2&0\\ 0&0&\lambda_3&0&0&\mu_3\\ \mu_1&0&0&\lambda_1&0&0\\ 0&\mu_2&0&0&\lambda_2&0\\ 0&0&\mu_3&0&0&\lambda_3\end{pmatrix}$$

relative to the basis $e_{1} ∧ e_{2}, e_{1} ∧ e_{3}, e_{1} ∧ e_{4}, e_{3} ∧ e_{4}, e_{4} ∧ e_{2}, e_{2} ∧ e_{3}$. One has that $μ_{1} + μ_{2} + μ_{3}$ is zero and that $λ_{1} + λ_{2} + λ_{3}$ is one-fourth of the scalar curvature of $g$ at $p$. Furthermore, under the conditions $λ_{1} ≤ λ_{2} ≤ λ_{3}$ and $μ_{1} ≤ μ_{2} ≤ μ_{3}$, 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
 * $$\begin{align}

\chi(M)&=\frac{1}{4\pi^2}\int_M\big(\lambda_1^2+\lambda_2^2+\lambda_3^2+\mu_1^2+\mu_2^2+\mu_3^2\big)\,d\mu_g\\ \tau(M)&=\frac{1}{3\pi^2}\int_M\big(\lambda_1\mu_1+\lambda_2\mu_2+\lambda_3\mu_3\big)\,d\mu_g. \end{align}$$ Equipped with these tools, the Hitchin-Thorpe inequality amounts to the elementary observation
 * $$\lambda_1^2+\lambda_2^2+\lambda_3^2+\mu_1^2+\mu_2^2+\mu_3^2=\underbrace{(\lambda_1-\mu_1)^2+(\lambda_2-\mu_2)^2+(\lambda_3-\mu_3)^2}_{\geq 0}+2\big(\lambda_1\mu_1+\lambda_2\mu_2+\lambda_3\mu_3\big).$$

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


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

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.