euler-form-of-S2.egi

;;; Parameters

(define $x [| θ φ |])

(define $X [|(* r (sin θ) (cos φ)) ; = x
             (* r (sin θ) (sin φ)) ; = y
             (* r (cos θ))         ; = z
             |])

;;; Local basis

(define $e ((flip ∂/∂) x~# X_#))
e
;[|[|(* r (cos θ) (cos φ)) (* r (cos θ) (sin φ)) (* -1 r (sin θ)) |]
;  [|(* -1 r (sin θ) (sin φ)) (* r (sin θ) (cos φ)) 0 |]
;  |]_#~#

;;; Metric tensor

(define $g__ (generate-tensor 2#(V.* e_%1 e_%2) {2 2}))
(define $g~~ (M.inverse g_#_#))

g_#_#;[| [| r^2 0 |] [| 0 (* r^2 (sin θ)^2) |] |]_#_#
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_#_#)))

;;; Connection form

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

(define $ω0 Γ~#_#_#)
ω0~#_#_1;[| [| 0 0 |] [| 0 (/ (cos θ) (sin θ)) |] |]~#_#
ω0~#_#_2;[| [| 0 (* -1 (sin θ) (cos θ)) |] [| (/ (cos θ) (sin θ)) 0 |] |]~#_#

(define $A [|[| (/ 1 r) 0 |] [| 0 (/ 1 (* r (sin θ))) |]|])

(define $ω (+ (. (M.inverse A)~i_j ω0~j_k A~k_l) (. (M.inverse A)~i_j (d A~j_l))))
ω~#_#_1;[| [| 0 0 |] [| 0 0 |] |]~#_#
ω~#_#_2;[| [| 0 (* -1 (cos θ)) |] [| (cos θ) 0 |] |]~#_#

;;; Curvature form

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

(define $Ω
  (with-symbols {i j}
    (df-normalize (+ (d ω~i_j)
                     (wedge ω~i_k ω~k_j)))))
Ω~#_#_1_2;[| [| 0 (sin θ) |] [| (* -1 (sin θ)) 0 |] |]~#_#
Ω~#_#_2_1;[| [| 0 (* -1 (sin θ)) |] [| (sin θ) 0 |] |]~#_#
Ω~1_2;[| [| 0 (sin θ) |] [| (* -1 (sin θ)) 0 |] |]
Ω~2_1;[| [| 0 (* -1 (sin θ)) |] [| (sin θ) 0 |] |]

;;; Euler form

(define $euler-form (* (/ 1 (* 2 π)) (- Ω~1_2 Ω~2_1)))

euler-form;[| [| 0 (/ (sin θ) (* 2 π)) |] [| (/ (* -1 (sin θ)) (* 2 π)) 0 |] |]

; χ(S^2) = ∫ dθ dφ (/ (sin θ) (* 2 π)) = ∫ dθ (sin θ)
; = [ (* -1 (cos θ)) ] 0-π = (cos 0) - (cos π) = 2