Scalar curvature

The scalar curvature (or the Ricci scalar) is the simplest invariant of a pseudo-Riemannian manifold. It assigns a single real number to every point on the manifold, representing the amount by which the volume of a small geodesic ball deviates from that of a standard ball in Euclidean space.

Given a pseudo-Riemannian manifold $(M, g)$ with Ricci curvature tensor with components $R_{a b}$ , the scalar curvature is the number

R = g^{a b} R_{a b} .

Interpretation

From this paper.
It turns out the scalar curvature has a meaning very similar to the Gaussian curvature. If we imagine instead of taking a circle, taking a generalized $D - 1$ sphere, i.e. the set of all points a geodesic distance $ϵ$ from a given starting point $x_{0}^{μ}$ . We can calculate the area of this sphere in flat space, but in curved space the area will deviate from the one we calculated by an amount proportional to the curvature. Thus, we find that the scalar curvature is

R = lim_{ϵ \to 0} \frac{6 D}{ϵ^{2}} [1 - \frac{A_{c u r v e d} (ϵ)}{A_{f l a t} (ϵ)}] .

Computation via the Riemann Tensor

In terms of the fully covariant Riemann curvature tensor $R_{a b c d}$ , the expression is:

R = g^{a b} g^{c d} R_{a c b d}

2-Dimensional Riemannian Manifolds

In two dimensions ( $n = 2$ ), the Riemann tensor has only one independent component. This leads to several significant simplifications: for a 2D surface, the scalar curvature $R$ is exactly twice the Gaussian curvature $K$ :

R = 2 K

Why?
The scalar curvature $R$ is defined as the trace of the Ricci tensor, which itself is a trace of the Riemann tensor. Geometrically, $R$ can be thought of as the sum of all sectional curvature $κ (Π)$ for an orthonormal basis.

In an $n$ -dimensional manifold, the scalar curvature is:

R = \sum_{i \neq j} sec (e_{i}, e_{j})

But in 2D, there is only one possible plane (the tangent plane itself), so the sum only has two terms for the indices $(1, 2)$ and $(2, 1)$ . Since $sec (e_{1}, e_{2}) = sec (e_{2}, e_{1}) = K$ , the sum becomes:

R = K + K = 2 K

Algebraically
If we look at the components of the Riemann tensor in 2D, the only non-vanishing, independent component is $R_{1212}$ . The symmetries of the tensor require that:

R_{1212} = - R_{2112} = - R_{1221} = R_{2121}

The Gaussian curvature $K$ is shown to be (see Riemann curvature tensor#Relation to Gaussian curvature of surfaces):

K = \frac{R_{1212}}{g}

(where $g$ is the determinant of the metric).

When we calculate the Scalar curvature $R = g^{a b} R_{a b}$ , we are effectively summing these components. In a 2D orthonormal frame ( $g_{a b} = δ_{a b}$ ), this expands to:

R = R_{11} + R_{22}

R = (R^{1}_{111} + R^{2}_{121}) + (R^{1}_{212} + R^{2}_{222})

Due to antisymmetry, $R^{1}_{111}$ and $R^{2}_{222}$ are zero. We are left with:

R = R_{2121} + R_{1212} = K + K = 2 K