# Curvature

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Informally, **curvature** is the obstruction to any geometric object being flat. In the case of connections , this obstruction is measured by the **curvature tensor**

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**

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**.