Hitchin–Thorpe inequality explained

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 (3)/(2) |\tau(M)|, where is the Euler characteristic of and is the signature of .

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 is an Einstein manifold for which equality in the Hitchin-Thorpe inequality is obtained, then the Ricci curvature of is zero; if the sectional curvature is not identically equal to zero, then 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 be a four-dimensional smooth Riemannian manifold which is Einstein. Given any point of, there exists a -orthonormal basis of the tangent space such that the curvature operator, which is a symmetric linear map of into itself, has matrix

\begin{pmatrix} λ _{1&0&0&\mu} _{1&0&0\ 0&λ} _{2&0&0&\mu} _{2&0\ 0&0&λ} _{3&0&0&\mu} _{3\ \mu} _{1&0&0&λ} _{1&0&0\ 0&\mu} _{2&0&0&λ} _{2&0\ 0&0&\mu} _{3&0&0&λ} _{3\end{pmatrix}} relative to the basis . One has that is zero and that is one-fourth of the scalar curvature of at . Furthermore, under the conditions and, each of these six functions is uniquely determined and defines a continuous real-valued function on .

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

\begin{align} \chi(M)& = (1)/(4 \pi ^{2}) \int _{M(λ} ^{2 ) d\mu} ^{g\\ \tau(M)& = (1)/(3 \pi ^{2})} \int _{M(λ} _{1\mu} _{1+λ} _{2\mu} _{2+λ} _{3\mu} _{3)d\mu} _{g. \end{align}} Equipped with these tools, the Hitchin-Thorpe inequality amounts to the elementary observationλ ^{2= \underbrace{( λ} _{1- \mu} ^{2+ ( λ} ^{2- \mu} ^{2+ ( λ} ^{3- \mu} ^{2 }} ^{\geq 0} +2 ( λ _{1\mu} _{1+λ} _{2\mu} _{2+λ} _{3\mu} _{3).}

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 that carry no Einstein metrics but nevertheless satisfy

\chi(M) > (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. J. . Thorpe . Some remarks on the Gauss-Bonnet formula . J. Math. Mech. . 18 . 8 . 1969 . 779–786 . 24893137 .
2. N. . Hitchin . Compact four-dimensional Einstein manifolds . J. Diff. Geom. . 9 . 3 . 1974 . 435–442 . 10.4310/jdg/1214432419 . free .
3. Berger . Marcel . 1961 . Sur quelques variétés d'Einstein compactes . Annali di Matematica Pura ed Applicata . fr . 53 . 1 . 89–95 . 10.1007/BF02417787 . 117985766 . 0373-3114. free .
4. Book: Besse, Arthur L. . Einstein Manifolds . Springer . 1987 . 3-540-74120-8 . Classics in Mathematics . Berlin . registration.
5. C. . LeBrun . Four-Manifolds without Einstein Metrics . Math. Res. Lett. . 3 . 2 . 1996 . 133–147 . 10.4310/MRL.1996.v3.n2.a1 . free .
6. A. . Sambusetti . An obstruction to the existence of Einstein metrics on 4-manifolds . C. R. Acad. Sci. Paris . 322 . 12 . 1996 . 1213–1218 . 0764-4442 .

References

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