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

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

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.[1] Nigel Hitchin then rediscovered the inequality, and gave a complete characterization of the equality case in 1974;[2] 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.[3][4]

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

${\displaystyle {\begin{pmatrix}\lambda _{1}&0&0&\mu _{1}&0&0\\0&\lambda _{2}&0&0&\mu _{2}&0\\0&0&\lambda _{3}&0&0&\mu _{3}\\\mu _{1}&0&0&\lambda _{1}&0&0\\0&\mu _{2}&0&0&\lambda _{2}&0\\0&0&\mu _{3}&0&0&\lambda _{3}\end{pmatrix}}}$

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

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

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

${\displaystyle \lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}+\mu _{1}^{2}+\mu _{2}^{2}+\mu _{3}^{2}=\underbrace {(\lambda _{1}-\mu _{1})^{2}+(\lambda _{2}-\mu _{2})^{2}+(\lambda _{3}-\mu _{3})^{2}} _{\geq 0}+2{\big (}\lambda _{1}\mu _{1}+\lambda _{2}\mu _{2}+\lambda _{3}\mu _{3}{\big )}.}$

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.[5] 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.[6]

Footnotes

1. Thorpe, J. (1969). "Some remarks on the Gauss-Bonnet formula". J. Math. Mech. 18 (8): 779–786. JSTOR 24893137.
2. Hitchin, N. (1974). "Compact four-dimensional Einstein manifolds". J. Diff. Geom. 9 (3): 435–442. doi:10.4310/jdg/1214432419.
3. 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.
4. Besse, Arthur L. (1987). Einstein Manifolds. Classics in Mathematics. Berlin: Springer. ISBN 3-540-74120-8.
5. LeBrun, C. (1996). "Four-Manifolds without Einstein Metrics". Math. Res. Lett. 3 (2): 133–147. doi:10.4310/MRL.1996.v3.n2.a1.
6. 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.

References

• Besse, Arthur L. (1987). Einstein Manifolds. Classics in Mathematics. Berlin: Springer. ISBN 3-540-74120-8.