Английская Википедия: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>\chi(M) \geq \frac{3}{2}|\tau(M)|,</math>
where Шаблон:Math is the Euler characteristic of Шаблон:Mvar and Шаблон:Math is the signature of Шаблон:Mvar.
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 Шаблон:Math is an Einstein manifold for which equality in the Hitchin-Thorpe inequality is obtained, then the Ricci curvature of Шаблон:Mvar is zero; if the sectional curvature is not identically equal to zero, then Шаблон:Math is a Calabi–Yau manifold whose universal cover is a K3 surface.
Шаблон:AnchorAlready in 1961, Marcel Berger showed that the Euler characteristic is always non-negative.[3][4]
Proof
Let Шаблон:Math be a four-dimensional smooth Riemannian manifold which is Einstein. Given any point Шаблон:Mvar of Шаблон:Mvar, there exists a Шаблон:Math-orthonormal basis Шаблон:Math of the tangent space Шаблон:Math such that the curvature operator Шаблон:Math, which is a symmetric linear map of Шаблон:Math into itself, has matrix
- <math>\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 Шаблон:Math. One has that Шаблон:Math is zero and that Шаблон:Math is one-fourth of the scalar curvature of Шаблон:Mvar at Шаблон:Mvar. Furthermore, under the conditions Шаблон:Math and Шаблон:Math, each of these six functions is uniquely determined and defines a continuous real-valued function on Шаблон:Mvar.
According to Chern-Weil theory, if Шаблон:Mvar is oriented then the Euler characteristic and signature of Шаблон:Mvar can be computed by
- <math>\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>\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 Шаблон:Mvar that carry no Einstein metrics but nevertheless satisfy
- <math>\chi(M) > \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.[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
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