A Torsional Topological Invariant

Understand

Curvature and torsion are the two tensors characterizing a general Riemannian spacetime.

• 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 spacetime.
• 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.