Riemann Curvature Tensor of $S^2$¶

A round 2-sphere of radius $r$ has constant sectional curvature $1/r^2$. This notebook builds its metric, Levi-Civita connection, Riemann tensor, Ricci tensor, and scalar curvature in indexed Egison notation.

We use the convention

$$ R^i{}_{jkl} = \partial_k\Gamma^i{}_{jl} - \partial_l\Gamma^i{}_{jk} + \Gamma^m{}_{jl}\Gamma^i{}_{mk} - \Gamma^m{}_{jk}\Gamma^i{}_{ml}. $$

Hyperspherical chart¶

The coordinates are $x=(\theta, \phi)$. The standard embedding $X:S^2\hookrightarrow\mathbb{R}^3$ is built by successively multiplying by sines: $X_1=r\cos \theta$, $X_2=r\sin \theta\cos \phi$, and so on. It makes $X\mathbin{\cdot}X=r^2$ manifest.

The chart excludes the usual coordinate poles; those singularities are features of hyperspherical coordinates, not of the round geometry.

declare symbol r, θ, φ: MathValue

def x : Vector MathValue := [| θ, φ |]

def X : Vector MathValue := [| r * cos θ, r * sin θ * cos φ, r * sin θ * sin φ |]

$\begin{pmatrix} \cos(θ) r \\ \cos(φ) \sin(θ) r \\ \sin(θ) \sin(φ) r\\ \end{pmatrix}$

Metric and inverse metric¶

Differentiating the embedding gives an orthogonal coordinate basis. Its line element is

$$ ds^2=r^2\left(d\theta^2 + \sin^2 \theta d\phi^2\right). $$

Egison differentiates the embedding to obtain the coordinate tangent vectors and constructs every component as their dot product, $g_{ij}=\partial_iX\mathbin{\cdot}\partial_jX$. The inverse metric is then computed from that induced metric.

def e_i_j : Matrix MathValue := ∂/∂ X_j x~i

def g_i_j : Matrix MathValue :=
  generateTensor (\[a, b] -> V.* e_a_# e_b_#) [2, 2]

def g~i~j : Matrix MathValue := M.inverse g_#_#

g_#_#

$\begin{pmatrix} r^{2} & 0 \\ 0 & \sin(θ)^{2} r^{2} \\ \end{pmatrix}_{\#\#}^{\;\;}$

g~#~#

$\begin{pmatrix} r^{-2} & 0 \\ 0 & \sin(θ)^{-2} r^{-2} \\ \end{pmatrix}_{\;\;}^{\#\#}$

Levi-Civita connection¶

The Christoffel symbols of the first kind and second kind are

$$ \Gamma_{ijk}=\frac12 (\partial_jg_{ik}+\partial_kg_{ij}-\partial_ig_{jk}), \qquad \Gamma^i{}_{jk}=g^{im}\Gamma_{mjk}. $$

Repeated symbolic indices are contracted by .. The withSymbols block makes the dummy index local to the definition.

def Γ_i_j_k : Tensor MathValue :=
  (1 / 2) * (∂/∂ g_i_k x~j + ∂/∂ g_i_j x~k - ∂/∂ g_j_k x~i)

def Γ~i_j_k : Tensor MathValue := withSymbols [m]
  g~i~m . Γ_m_j_k

Γ~1_#_#

$\begin{pmatrix} 0 & 0 \\ 0 & -\cos(θ) \sin(θ) \\ \end{pmatrix}_{\#\#}^{\;\;}$

Γ~2_#_#

$\begin{pmatrix} 0 & \cos(θ) \sin(θ)^{-1} \\ \cos(θ) \sin(θ)^{-1} & 0 \\ \end{pmatrix}_{\#\#}^{\;\;}$

Riemann tensor¶

The following definition is a direct transcription of the stated convention. The two output cells sample the same coordinate two-plane with the first two tensor slots exchanged; their different coordinate factors are exactly what one expects in a non-orthonormal coordinate basis.

def R~i_j_k_l : Tensor MathValue := withSymbols [m]
  ∂/∂ Γ~i_j_l x~k - ∂/∂ Γ~i_j_k x~l
    + Γ~m_j_l . Γ~i_m_k - Γ~m_j_k . Γ~i_m_l

R~#_#_1_2

$\begin{pmatrix} 0 & \sin(θ)^{2} \\ -1 & 0 \\ \end{pmatrix}_{\;\#}^{\#\;}$

R~#_#_2_1

$\begin{pmatrix} 0 & -\sin(θ)^{2} \\ 1 & 0 \\ \end{pmatrix}_{\;\#}^{\#\;}$

Ricci and scalar curvature¶

Contracting the first and third Riemann indices gives

$$ \operatorname{Ric}_{ij}=R^m{}_{imj}, \qquad \mathcal{R}=g^{ij}\operatorname{Ric}_{ij}. $$

For a round $S^2$ the coordinate-free prediction is

$$ \operatorname{Ric}=\frac{1}{r^2}g, \qquad \mathcal{R}=\frac{2}{r^2}. $$

def Ric_i_j : Matrix MathValue := withSymbols [m]
  sum (contract R~m_i_m_j)

def scalarCurvature : MathValue := withSymbols [i, j]
  g~i~j . Ric_i_j

Ric_#_#

$\begin{pmatrix} 1 & 0 \\ 0 & \sin(θ)^{2} \\ \end{pmatrix}_{\#\#}^{\;\;}$

scalarCurvature

$2 r^{-2}$

Interpretation¶

The sampled components are coordinate dependent, but their contractions recover the invariant statement: every tangent two-plane has sectional curvature $1/r^2$, the metric is Einstein, and the scalar curvature is $2/r^2$. Factors such as $\sin \theta$ vanish at chart poles because the coordinate frame degenerates there; the curvature itself remains smooth.