Curvature

Informally, curvature is the obstruction to any geometric object being flat. In the case of connections $A_{\alpha }$, this obstruction is measured by the curvature tensor

$F_{\alpha \beta }=[D_{\alpha },D_{\beta }]=\partial _{\alpha }A_{\beta }-\partial _{\beta }A_{\alpha }+[A_{\alpha },A_{\beta }].$

If the gauge group is abelian, then the last term can be omitted.

For Riemannian or Lorentzian manifolds, the curvature of the Levi-Civita connection gives the Riemann curvature tensor

$\operatorname {Riem} _{\alpha \beta \gamma }^{\delta }X^{\gamma }=[\nabla _{\alpha },\nabla _{\beta }]X^{\delta }$

which can be expressed in terms of second and first derivatives of the metric. Contracting two of the indices of the Riemann curvature tensor yields the Ricci curvature tensor, which plays a prominent role in the Einstein equation. Contracting all four indices yields the Ricci scalar curvature.