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

: $\backslash chi(M)\; \backslash geq\; \backslash frac\{3\}\{2\}|\backslash tau(M)|,$

where {{math|χ(M)}} is the Euler characteristic of {{mvar|M}} and {{math|τ(M)}} is the signature of {{mvar|M}}.

This inequality was first stated by John Thorpe in a footnote to a 1969 paper focusing on manifolds of higher dimension.{{cite journal |first=J. |last=Thorpe |title=Some remarks on the Gauss-Bonnet formula |journal=J. Math. Mech. |volume=18 |issue=8 |year=1969 |pages=779–786 |jstor=24893137 }} Nigel Hitchin then rediscovered the inequality, and gave a complete characterization of the equality case in 1974;{{cite journal |first=N. |last=Hitchin |title=Compact four-dimensional Einstein manifolds |journal=J. Diff. Geom. |volume=9 |issue=3 |year=1974 |pages=435–442 |doi=10.4310/jdg/1214432419 |doi-access=free }} he found that if {{math|(M, g)}} is an Einstein manifold for which equality in the Hitchin-Thorpe inequality is obtained, then the Ricci curvature of {{mvar|g}} is zero; if the sectional curvature is not identically equal to zero, then {{math|(M, g)}} is a Calabi–Yau manifold whose universal cover is a K3 surface.

{{Anchor|Berger's inequality}}Already in 1961, Marcel Berger showed that the Euler characteristic is always non-negative.{{Cite journal |last=Berger |first=Marcel |date=1961 |title=Sur quelques variétés d'Einstein compactes |journal=Annali di Matematica Pura ed Applicata |language=fr |volume=53 |issue=1 |pages=89–95 |doi=10.1007/BF02417787 |s2cid=117985766 |issn=0373-3114|doi-access=free }}{{cite book |last=Besse |first=Arthur L. |url=https://archive.org/details/einsteinmanifold0000bess |title=Einstein Manifolds |publisher=Springer |year=1987 |isbn=3-540-74120-8 |series=Classics in Mathematics |location=Berlin |url-access=registration}}

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

:

relative to the basis {{math|e₁ ∧ e₂, e₁ ∧ e₃, e₁ ∧ e₄, e₃ ∧ e₄, e₄ ∧ e₂, e₂ ∧ e₃}}. One has that {{math|μ₁ + μ₂ + μ₃}} is zero and that {{math|λ₁ + λ₂ + λ₃}} is one-fourth of the scalar curvature of {{mvar|g}} at {{mvar|p}}. Furthermore, under the conditions {{math|λ₁ ≤ λ₂ ≤ λ₃}} and {{math|μ₁ ≤ μ₂ ≤ μ₃}}, each of these six functions is uniquely determined and defines a continuous real-valued function on {{mvar|M}}.

According to Chern-Weil theory, if {{mvar|M}} is oriented then the Euler characteristic and signature of {{mvar|M}} can be computed by

:$\backslash begin\{align\}$

\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

:$\backslash lambda\_1^2+\backslash lambda\_2^2+\backslash lambda\_3^2+\backslash mu\_1^2+\backslash mu\_2^2+\backslash mu\_3^2=\backslash underbrace\{(\backslash lambda\_1-\backslash mu\_1)^2+(\backslash lambda\_2-\backslash mu\_2)^2+(\backslash lambda\_3-\backslash mu\_3)^2\}\_\{\backslash geq\; 0\}+2\backslash big(\backslash lambda\_1\backslash mu\_1+\backslash lambda\_2\backslash mu\_2+\backslash lambda\_3\backslash mu\_3\backslash big).$

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 {{mvar|M}} that carry no Einstein metrics but nevertheless satisfy

: $\backslash chi(M)\; >\; \backslash frac\{3\}\{2\}|\backslash tau(M)|.$

LeBrun's examples are actually simply connected, and the relevant obstruction depends on the smooth structure of the manifold.{{cite journal |first=C. |last=LeBrun |title=Four-Manifolds without Einstein Metrics |journal=Math. Res. Lett. |volume=3 |issue=2 |year=1996 |pages=133–147 |doi=10.4310/MRL.1996.v3.n2.a1 |doi-access=free }} 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.{{cite journal |first=A. |last=Sambusetti |title=An obstruction to the existence of Einstein metrics on 4-manifolds |journal=C. R. Acad. Sci. Paris |volume=322 |issue=12 |year=1996 |pages=1213–1218 |issn=0764-4442 }}

• {{cite book | first = Arthur L. | last = Besse | title = Einstein Manifolds | url = https://archive.org/details/einsteinmanifold0000bess | url-access = registration | series = Classics in Mathematics | publisher = Springer | location = Berlin | year = 1987 | isbn = 3-540-74120-8}}

{{DEFAULTSORT:Hitchin-Thorpe inequality}}

Einstein manifolds

Category:Geometric inequalities

Einstein manifold