Ricci scalar curvature

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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 $g_{munu}$ (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:

${displaystyle R=-g^{mu nu }left[Gamma _{mu nu }^{lambda }Gamma _{lambda sigma }^{sigma }-Gamma _{mu sigma }^{lambda }Gamma _{nu lambda }^{sigma }right]-partial _{nu }left[g^{mu nu }Gamma _{mu sigma }^{sigma }-g^{mu sigma }Gamma _{mu sigma }^{nu }right]}$

Where the Christoffel symbols that appear in the previous expression are calculated from the first derivatives of the components of the metric tensor:

{displaystyle Gamma _{kl}^{i}={frac {1}{2}}g^{im}left({frac {partial g_{mk}}{partial x^{l}}}+{frac {partial g_{ml}}{partial x^{k}}}-{frac {partial g_{kl}}{partial x^{m}}}right)}

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