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Hitchin–Thorpe inequality

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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

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Let M be a closed, oriented, four-dimensional smooth manifold. If there exists a Riemannian metric on M which is an Einstein metric, then

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

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Let (M, g) be a four-dimensional smooth Riemannian manifold which is Einstein. Given any point p of M, there exists a gp-orthonormal basis e1, e2, e3, e4 of the tangent space TpM such that the curvature operator Rmp, which is a symmetric linear map of 2TpM into itself, has matrix

relative to the basis e1e2, e1e3, e1e4, e3e4, e4e2, e2e3. 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

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

Failure of the converse

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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

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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  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

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