In differential geometry the Hitchin–Thorpe inequality is a famous relation which restricts the topology of 4-manifold

4-manifold

In mathematics, 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different...

s that carry an Einstein metric

Einstein manifold

In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian manifold whose Ricci tensor is proportional to the 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

Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...

 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. Nigel Hitchin

Nigel Hitchin

Nigel Hitchin is a British mathematician working in the fields of differential geometry, algebraic geometry, and mathematical physics.-Academic career:...

 then rediscovered the inequality, and gave a complete characterization 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

Ricci decomposition

In semi-Riemannian geometry, the Ricci decomposition is a way of breaking up the Riemann curvature tensor of a pseudo-Riemannian manifold into pieces with useful individual algebraic properties...

 of the Riemann curvature tensor

Riemann curvature tensor

In the mathematical field of differential geometry, the Riemann curvature tensor, or Riemann–Christoffel tensor after Bernhard Riemann and Elwin Bruno Christoffel, is the most standard way to express curvature of Riemannian manifolds...

 and the Generalized Gauss-Bonnet theorem

Generalized Gauss-Bonnet theorem

In mathematics, the generalized Gauss–Bonnet theorem presents the Euler characteristic of a closed even-dimensional Riemannian manifold as an integral of a certain polynomial derived from its curvature...

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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 are actually simply connected, and the relevant obstruction depends on the smooth structure of the manifold. 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.