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.^[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 $e 1, e 2, e 3, e 4$ of the tangent space $T p M$ such that the curvature operator $Rm p$ , which is a symmetric linear map of $\land 2 T p M$ 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 $e 1 \land e 2, e 1 \land e 3, e 1 \land e 4, e 3 \land e 4, e 4 \land e 2, e 2 \land e 3$ . 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 $λ 1 \leq λ 2 \leq λ 3$ and $μ 1 \leq μ 2 \leq μ 3$ , 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.^[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

^ 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.

References

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

[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.

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