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In [[differential geometry]] the '''Hitchin–Thorpe inequality''' is a famous relation which restricts the topology of [[4-manifold]]s that carry an [[Einstein manifold|Einstein metric]]. | |||
== Statement of the Hitchin–Thorpe inequality == | |||
Let ''M'' be a compact, oriented, smooth four-dimensional manifold. If there exists a [[Riemannian metric]] on ''M'' which is an [[Einstein metric]], then following inequality holds | |||
: <math>\chi(M) \geq \frac{3}{2}|\tau(M)|,</math> | |||
where <math>\chi(M)</math> is the [[Euler characteristic]] of <math>M</math> and <math>\tau(M)</math> is the [[signature (topology)|signature]] of <math>M</math>. This inequality was first stated by John Thorpe<ref>J. Thorpe, ''Some remarks on on the Gauss-Bonnet formula'', J. Math. Mech. 18 (1969) pp. 779--786.</ref> in a footnote to a 1969 paper focusing | |||
on manifolds of higher dimension. [[Nigel Hitchin]] then rediscovered the inequality, and gave a complete characterization <ref>N. Hitchin, ''On compact four-dimensional Einstein manifolds'', J. Diff. Geom. 9 (1974) pp. 435--442.</ref> of the equality case in 1974; he found that if <math>(M,g)</math> is an Einstein manifold with <math>\chi(M) = \frac{3}{2}|\tau(M)|,</math> then <math>(M,g)</math> must be a flat torus, a [[Calabi–Yau manifold]], or a quotient thereof. | |||
== Idea of the proof == | |||
The main ingredients in the proof of the Hitchin–Thorpe inequality are the [[Ricci decomposition|decomposition]] of the [[Riemann curvature tensor]] and the [[Generalized Gauss-Bonnet theorem]]. | |||
== 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 | |||
: <math>\chi(M) > \frac{3}{2}|\tau(M)|.</math> | |||
LeBrun's examples <ref>[[Claude LeBrun|C. LeBrun]], ''Four-manifolds without Einstein Metrics'', Math. Res. Letters 3 (1996) pp. 133--147.</ref> are actually simply connected, and the relevant obstruction depends on the smooth structure of the manifold. By contrast, Sambusetti's obstruction <ref>A. Sambusetti, ''An obstruction to the existence of Einstein metrics on 4-manifolds'', C.R. Acad. Sci. Paris 322 (1996) pp. 1213--1218.</ref> 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. | |||
==Footnotes== | |||
<references/> | |||
== References == | |||
*{{cite book | first = Arthur L. | last = Besse | title = Einstein Manifolds | series = Classics in Mathematics | publisher = Springer | location = Berlin | year = 1987 | isbn = 3-540-74120-8}} | |||
{{DEFAULTSORT:Hitchin-Thorpe inequality}} | |||
[[Category:Riemannian manifolds|Einstein manifolds]] | |||
[[Category:Geometric inequalities]] | |||
[[Category:4-manifolds|Einstein manifold]] | |||
Latest revision as of 09:44, 20 August 2013
In differential geometry the Hitchin–Thorpe inequality is a famous relation which restricts the topology of 4-manifolds that carry an Einstein metric.
Statement of the Hitchin–Thorpe inequality
Let M be a compact, oriented, smooth four-dimensional manifold. If there exists a Riemannian metric on M which is an Einstein metric, then following inequality holds
where is the Euler characteristic of and is the signature of . This inequality was first stated by John Thorpe[1] in a footnote to a 1969 paper focusing on manifolds of higher dimension. Nigel Hitchin then rediscovered the inequality, and gave a complete characterization [2] of the equality case in 1974; he found that if is an Einstein manifold with then must be a flat torus, a Calabi–Yau manifold, or a quotient thereof.
Idea of the proof
The main ingredients in the proof of the Hitchin–Thorpe inequality are the decomposition of the Riemann curvature tensor and the Generalized Gauss-Bonnet theorem.
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
LeBrun's examples [3] are actually simply connected, and the relevant obstruction depends on the smooth structure of the manifold. By contrast, Sambusetti's obstruction [4] 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.
Footnotes
- ↑ J. Thorpe, Some remarks on on the Gauss-Bonnet formula, J. Math. Mech. 18 (1969) pp. 779--786.
- ↑ N. Hitchin, On compact four-dimensional Einstein manifolds, J. Diff. Geom. 9 (1974) pp. 435--442.
- ↑ C. LeBrun, Four-manifolds without Einstein Metrics, Math. Res. Letters 3 (1996) pp. 133--147.
- ↑ A. Sambusetti, An obstruction to the existence of Einstein metrics on 4-manifolds, C.R. Acad. Sci. Paris 322 (1996) pp. 1213--1218.
References
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