nLab scalar curvature

Definition

For $(X,e)$ a (pseudo-)Riemannian manifold with smooth manifold $X$ and vielbein field $e$, its scalar curvature is the smooth function

$$R(e) \in C^\infty(X, \mathbb{R})$$

defined to be the trace of the Ricci tensor of $e$

$$R(e) \coloneqq tr_e Ric(c) \,.$$

Related concepts

• scalar

• The product of the scalar curvature with the volume form is the Lagrangian of the theory (physics) of gravity. The corresponding action functional is the Einstein-Hilbert action.

• Lichnerowicz formula

curvature in Riemannian geometry
Riemann curvature
Ricci curvature
scalar curvature
sectional curvature
p-curvature