We derive a new Gauss–Bonnet type identity in Riemann-Cartan geometry: (−g)1/2εμνλρ (Rμνλρ + (1/2) CαμνCαλν) = ∂μ (−(−g)1/2εμνλρCμνλρ), where Cαμν is the torsion tensor.

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Curvature and torsion are the two tensors characterizing a general Riemannian space-time. In Einstein's General Theory of Gravitation, with torsion postulated to vanish and the affine connection identified to the Christoffel symbol, only the curvature tensor plays the central role. For such a purely metric geometry, two well-known topological invariants, namely the Euler class and the Pontryagin class, are useful in characterizing the topological properties of the space-time. From a gauge theory point of view, and especially in the presence of spin, torsion naturally comes into play, and the underlying spacetime is no longer purely metric. We describe a torsional topological invariant, discovered in 1982, that has now found increasing usefulness in recent developments.