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

[math]\displaystyle{ \chi(M) \geq \frac{3}{2}|\tau(M)|, }[/math]

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 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

[math]\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} }[/math]

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

[math]\displaystyle{ \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} }[/math]

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

[math]\displaystyle{ \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). }[/math]

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

[math]\displaystyle{ \chi(M) \gt \frac{3}{2}|\tau(M)|. }[/math]

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.

References

• Besse, Arthur L. (1987). Einstein Manifolds. Classics in Mathematics. Berlin: Springer. ISBN 3-540-74120-8. https://archive.org/details/einsteinmanifold0000bess.