scalar curvature (changes) in nLab

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Definition

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

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

defined to be the trace of the Ricci tensor of ee

R(e)≔tr eRic(c). R(e) \coloneqq tr_e Ric(c) \,.

Examples

\begin{example}\label{RicciTensorOfRoundNSphere} For n∈ℕ >0n \in \mathbb{N}_{\gt 0} and r∈ℝ >0r \in \mathbb{R}_{\gt 0}, the Ricci tensor of the round $n$-sphere S nS^n of radius rr satisfies

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

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

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

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

\end{example} (e.g. Lee 2018, Cor. 11.20)

• 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