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 $e$

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

Examples

Example

For $n \in \mathbb{N}_{\gt 0}$ and $r \in \mathbb{R}_{\gt 0}$, the Ricci tensor of the round $n$-sphere $S^n$ of radius $r$ satisfies

$$Ric(v,v) \;=\; \frac{n-1}{r^2}$$

for all unit-length tangent vectors $v \in T S^n$, ${\vert v \vert} = 1$.

Accordingly, the scalar curvature of the round $n$-sphere of radius $r$ is the constant function with value

$$\mathrm{R} \;=\; \frac{n(n-1)}{r^2} \,.$$

(e.g. Lee 2018, Cor. 11.20)

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

References

Most references listed at Riemannian geometry discuss scalar curvature, for instance

• John M. Lee, Riemannian manifolds. An introduction to curvature, Graduate Texts in Mathematics 176 Springer (1997) [ISBN: 0-387-98271-X]

  second edition (retitled):

  John M. Lee, Introduction to Riemannian Manifolds, Springer (2018) [ISBN:978-3-319-91754-2, doi:10.1007/978-3-319-91755-9]

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