Ricci scalar curvature
In mathematics, the scalar curvature of a surface is twice the familiar Gaussian curvature. For the highest dimensional Riemannian manifolds (n > 2), it is twice the sum of all sectional curvatures along all 2-planes traversed by a certain orthonormal frame. Mathematically, the scalar curvature or curvature scalar, which is usually designated by the letters R or S, also coincides with the total trace of the Ricci curvature as well as the curvature tensor.
Expression in components
The Ricci curvature climber R can be easily expressed in terms of metric tensor gμ μ .. {displaystyle g_{mu nu }(and its first derivatives) that defines the geometry of the riemannian surface or variety whose curvature scale we intend to find, using Einstein's summing agreement:
R=− − gμ μ .. [chuckles]Interpreter Interpreter μ μ .. λ λ Interpreter Interpreter λ λ σ σ σ σ − − Interpreter Interpreter μ μ σ σ λ λ Interpreter Interpreter .. λ λ σ σ ]− − ▪ ▪ .. [chuckles]gμ μ .. Interpreter Interpreter μ μ σ σ σ σ − − gμ μ σ σ Interpreter Interpreter μ μ σ σ .. ]♪ Gamma ♪
![{displaystyle R=-g^{mu nu }left[Gamma _{mu nu }^{lambda }Gamma _{lambda sigma }^{sigma }-Gamma _{mu sigma }^{lambda }Gamma _{nu lambda }^{sigma }right]-partial _{nu }left[g^{mu nu }Gamma _{mu sigma }^{sigma }-g^{mu sigma }Gamma _{mu sigma }^{nu }right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132b079bf518c07df61573dccb6a610cf33a12cb)
Where the Christoffel symbols that appear in the previous expression are calculated from the first derivatives of the components of the metric tensor:
- Interpreter Interpreter kli=12gim(▪ ▪ gmk▪ ▪ xl+▪ ▪ gml▪ ▪ xk− − ▪ ▪ gkl▪ ▪ xm){displaystyle Gamma _{kl}^{i}={frac {1}{2}{2}{im}left({frac {partial g_{mk}}{partial x{l}}}}}}{frac {partial g_{ml}{partial x{k}{partial}}{
