# 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
    ${\begin{aligned}\chi (M)&={\frac {1}{4\pi ^{2}}}\int _{M}{\big\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}+\mu _{1}^{2}+\mu _{2}^{2}+\mu _{3}^{2}{\big )}\,d\mu _{g}\\\tau (M)&={\frac {1}{3\pi ^{2}}}\int _{M}{\big\lambda _{1}\mu _{1}+\lambda _{2}\mu _{2}+\lambda _{3}\mu _{3}{\big )}\,d\mu _{g}.\end{aligned}}$
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.

## 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.
* ↑ Berger, Marcel (1961). "Sur quelques variétés d'Einstein compactes". Annali di Matematica Pura ed Applicata (in French). 53 (1): 89–95. doi:10.1007/BF02417787. ISSN 0373-3114. S2CID 117985766.
* ↑ Besse, Arthur L. (1987). Einstein Manifolds. Classics in Mathematics. Berlin: Springer. ISBN 3-540-74120-8.
* ↑ LeBrun, C. (1996). "Four-Manifolds without Einstein Metrics". Math. Res. Lett. 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.