In differential geometry the Hitchin–Thorpe inequality is a relation which restricts the topology of 4-manifolds that carry an Einstein metric.
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.
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 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.