;;; Parameters and Metric tensor

(define $x [| θ φ |])

(define $g__ [| [| r^2 0 |] [| 0 (* r^2 (sin θ)^2) |] |])
(define $g~~ [| [| (/ 1 r^2) 0 |] [| 0 (/ 1 (* r^2 (sin θ)^2)) |] |])

;;; Christoffel symbols

(define $Γ_j_k_l
  (* (/ 1 2)
     (+ (∂/∂ g_j_l x_k)
        (∂/∂ g_j_k x_l)
        (* -1 (∂/∂ g_k_l x_j)))))

(define $Γ~__ (with-symbols {i} (. g~#~i Γ_i_#_#)))

;;; Riemann curvature tensor

(define $R~i_j_k_l
  (with-symbols {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_1;[| [| 0 0 |] [| 0 0 |] |]~#_#
R~#_#_1_2;[| [| 0 (sin θ)^2 |] [| -1 0 |] |]~#_#
R~#_#_2_1;[| [| 0 (* -1 (sin θ)^2) |] [| 1 0 |] |]~#_#
R~#_#_2_2;[| [| 0 0 |] [| 0 0 |] |]~#_#

;;; Connection form

(define $ω Γ~#_#_#)

;;; Curvature form

(define $wedge
  (lambda [%X %Y]
    !(. X Y)))

(define $d
  (lambda [%A]
    !((flip ∂/∂) x A)))

(define $D
  (lambda [%A]
    (with-symbols {i j}
      (+ (d A) (wedge ω~i_j A)))))

(define $Ω
  (with-symbols {i j}
    (df-normalize (+ (d ω~i_j)
                     (wedge ω~i_k ω~k_j)))))

Ω~#_#_1_1;[| [| 0 0 |] [| 0 0 |] |]~#_#
Ω~#_#_1_2;[| [| 0 (/ (sin θ)^2 2) |] [| (/ -1 2) 0 |] |]~#_#
Ω~#_#_2_1;[| [| 0 (/ (* -1 (sin θ)^2) 2) |] [| (/ 1 2) 0 |] |]~#_#
Ω~#_#_2_2;[| [| 0 0 |] [| 0 0 |] |]~#_#