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 where is the Euler characteristic 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 then rediscovered the inequality, and gave a complete characterization of the  equality case in 1974; he found that if is an Einstein manifold for which equality in the Hitchin-Thorpe inequality is obtained, then the Ricci curvature of is zero; if the sectional curvature is not identically equal to zero, then is a Calabi–Yau manifold whose universal cover is a K3 surface. Proof. Let be a four-dimensional smooth Riemannian manifold which is Einstein. Given any point of , there exists a -orthonormal basis of the tangent space such that the curvature operator , which is a symmetric linear map of into itself, has matrix relative to the basis . One has that  is zero and that is one-fourth of the scalar curvature of at . Furthermore, under the conditions and , each of these six functions is uniquely determined and defines a continuous real-valued function on . According to Chern-Weil theory, if is oriented then the Euler characteristic and signature of can be computed by Equipped with these tools, the Hitchin-Thorpe inequality amounts to the elementary observation 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 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.