nLab Hitchin-Thorpe inequality

Idea

The Hitchin-Thorpe inequality states the Euler characteristic is larger than one and a half times the absolute signature for a smooth orientable closed Einstein 4-manifold. It therefore serves as an obstruction for the existence of Einstein metrics. It can be proven using Chern-Weil theory.

Formulation

For a smooth orientable closed Einstein 4-manifold $M$, hence with $Ric=\lambda g$ for the Ricci curvature $Ric$ of its Riemannian metric $g$ and a scalar $\lambda\in\mathbb{R}$, one has:

$$\chi(M) \geq\frac{3}{2}|\sigma(M)|.$$

References

• John Thorpe, Some remarks on the Gauss-Bonnet formula, J. Math. Mech. 18 (8), pp. 779–786 (1969) [JSTOR:24893137]

• Nigel Hitchin, Compact four-dimensional Einstein manifolds, J. Diff. Geom. 9 (3), pp. 435–442 (1974) [doi:10.4310/jdg/1214432419]

See also:

• Wikipedia, Hitchin–Thorpe inequality