Curvature Two-Form on the Sphere

The Riemann tensor can be computed directly from Christoffel symbols or assembled as the curvature of the connection one-form. Cartan's second structure equation is

$$\Omega^i{}_j=d\omega^i{}_j+\omega^i{}_k\wedge\omega^k{}_j,$$

with $\Omega^i{}_j=\tfrac12R^i{}_{jkl}\,dx^k\wedge dx^l$.

Round metric on $S^2$

For a sphere of radius $r$ with coordinates $(\theta,\phi)$,

$$g_{ij}=\operatorname{diag}(r^2,r^2\sin^2\theta).$$

declare symbol r, θ, φ : MathValue

def x : Vector MathValue := [| θ, φ |]
def g_i_j : Matrix MathValue :=
  [| [| r ^ 2, 0 |], [| 0, r ^ 2 * (sin θ) ^ 2 |] |]_i_j
def g~i~j : Matrix MathValue :=
  [| [| 1 / r ^ 2, 0 |]
   , [| 0, 1 / (r ^ 2 * (sin θ) ^ 2) |] |]~i~j

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

Direct Riemann tensor

The Levi-Civita connection follows from metric compatibility and zero torsion. Egison's repeated symbolic indices perform the contraction over $m$.

def Γ_j_l_k : Tensor MathValue :=
  (1 / 2) *
    (∂/∂ g_j_l x~k + ∂/∂ g_j_k x~l - ∂/∂ g_k_l x~j)

def Γ~i_k_l : Tensor MathValue :=
  withSymbols [j] g~i~j . Γ_j_l_k

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

Cartan's curvature form

Regard the last Christoffel index as the one-form index. The exterior derivative adds one form index, and antisymmetrize projects the result onto a genuine two-form.

def d (t : Tensor MathValue) : Tensor MathValue :=
  !(flip ∂/∂) x t

def ω~i_j : Matrix MathValue := Γ~i_j_#

def Ω~i_j : Tensor MathValue := withSymbols [k]
  antisymmetrize (d ω~i_j + ω~i_k ∧ ω~k_j)

Ω~#_#_1_2
$\begin{pmatrix} 0 & \frac{1}{2} \sin(θ)^{2} \\ \frac{-1}{2} & 0 \\ \end{pmatrix}_{\;\#}^{\#\;}$

The curvature-form component is one half of the corresponding direct Riemann component, exactly as $\Omega^i{}_j=\tfrac12R^i{}_{jkl}dx^k\wedge dx^l$ requires. The two computational paths therefore agree while exposing different geometry: the first uses coordinate indices, and the second treats curvature as the field strength of a connection.

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