riemann-curvature-tensor-of-T2.egi

;;;
;;; Coordinates for Torus
;;;

(define $x [|θ φ|])

(define $X [|(* '(+ (* a (cos θ)) b) (cos φ)) ; = x
             (* '(+ (* a (cos θ)) b) (sin φ)) ; = y
             (* a (sin θ))                    ; = z
             |])

;;
;; Local basis
;;

(define $e ((flip ∂/∂) x~# X_#))
(assert-equal "Local basis"
  e_#_#
  [|[| (* -1 a (sin θ) (cos φ)) (* -1 a (sin θ) (sin φ)) (* a (cos θ)) |]
    [| (* -1 '(+ (* a (cos θ)) b) (sin φ)) (* '(+ (* a (cos θ)) b) (cos φ)) 0 |]
    |]_#_#)

;;
;; Metric tensor
;;

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

(assert-equal "Metric tensor 1" g_#_# [| [| a^2 0 |] [| 0 '(+ (* a (cos θ)) b)^2 |] |]_#_#)
(assert-equal "Metroc tensor 2" g~#~# [| [| (/ 1 a^2) 0 |] [| 0 (/ 1 '(+ (* a (cos θ)) b)^2) |] |]~#~#)

;;
;; Christoffel symbols of the first kind
;;

(define $Γ_i_j_k
  (* (/ 1 2)
     (+ (∂/∂ g_i_j x~k)
        (∂/∂ g_i_k x~j)
        (* -1 (∂/∂ g_j_k x~i)))))

(assert-equal "Christoffel symbols of the first kind" Γ_#_#_# (tensor {2 2 2} {0 0 0 (* '(+ (* a (cos θ)) b) a (sin θ)) 0 (* -1 '(+ (* a (cos θ)) b) a (sin θ)) (* -1 '(+ (* a (cos θ)) b) a (sin θ)) 0} )_#_#_#)
(assert-equal "Christoffel symbols of the first kind" Γ_1_#_# [| [| 0 0 |] [| 0 (* '(+ (* a (cos θ)) b) a (sin θ)) |] |]_#_#)
(assert-equal "Christoffel symbols of the first kind" Γ_2_#_# [| [| 0 (* -1 '(+ (* a (cos θ)) b) a (sin θ)) |] [| (* -1 '(+ (* a (cos θ)) b) a (sin θ)) 0 |] |]_#_#)

;;
;; Christoffel symbols of the second kind
;;

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

(assert-equal "Christoffel symbols of the second kind" Γ~#_#_# (tensor {2 2 2} {0 0 0 (/ (* '(+ (* a (cos θ)) b) (sin θ)) a) 0 (/ (* -1 a (sin θ)) '(+ (* a (cos θ)) b)) (/ (* -1 a (sin θ)) '(+ (* a (cos θ)) b)) 0} )~#_#_#)
(assert-equal "Christoffel symbols of the second kind" Γ~1_#_# [| [| 0 0 |] [| 0 (/ (* '(+ (* a (cos θ)) b) (sin θ)) a) |] |]_#_#)
(assert-equal "Christoffel symbols of the second kind" Γ~2_#_# [| [| 0 (/ (* -1 a (sin θ)) '(+ (* a (cos θ)) b)) |] [| (/ (* -1 a (sin θ)) '(+ (* a (cos θ)) b)) 0 |] |]_#_#)

;;
;; Covariant derivative of metric tensor
;;
(define $∇g___
  (with-symbols {i j m n}
    (- (∂/∂ g_i_j x~m)
       (. Γ~n_m_i g_n_j)
       (. Γ~n_m_j g_i_n))))

(assert-equal "Covariant derivative of metric tensor" ∇g_#_#_# (tensor {2 2 2} {0 0 0 0 0 0 0 0} ))

;;
;; 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)))))

(assert-equal "Riemann curvature" R~#_#_#_# (tensor {2 2 2 2} {0 0 0 0 0 (/ (* '(+ (* a (cos θ)) b) (cos θ)) a) (/ (* -1 '(+ (* a (cos θ)) b) (cos θ)) a) 0 0 (/ (* -1 a (cos θ)) '(+ (* a (cos θ)) b)) (/ (* a (cos θ)) '(+ (* a (cos θ)) b)) 0 0 0 0 0} )~#_#_#_#)
(assert-equal "Riemann curvature" R~#_#_1_1 [| [| 0 0 |] [| 0 0 |] |]~#_#)
(assert-equal "Riemann curvature" R~#_#_1_2 [| [| 0 (/ (* '(+ (* a (cos θ)) b) (cos θ)) a) |] [| (/ (* -1 a (cos θ)) '(+ (* a (cos θ)) b)) 0 |] |]~#_#)
(assert-equal "Riemann curvature" R~#_#_2_1 [| [| 0 (/ (* -1 '(+ (* a (cos θ)) b) (cos θ)) a) |] [| (/ (* a (cos θ)) '(+ (* a (cos θ)) b)) 0 |] |]~#_#)
(assert-equal "Riemann curvature" R~#_#_2_2 [| [| 0 0 |] [| 0 0 |] |]~#_#)

(define $R____ (with-symbols {i} (. g_i_# R~i_#_#_#)))

(assert-equal "Riemann curvature" R_#_#_#_# (tensor {2 2 2 2} {0 0 0 0 0 (* a '(+ (* a (cos θ)) b) (cos θ)) (* -1 a '(+ (* a (cos θ)) b) (cos θ)) 0 0 (* -1 '(+ (* a (cos θ)) b) a (cos θ)) (* '(+ (* a (cos θ)) b) a (cos θ)) 0 0 0 0 0} )_#_#_#_#)
(assert-equal "Riemann curvature" R_#_#_1_1 [| [| 0 0 |] [| 0 0 |] |]_#_#)
(assert-equal "Riemann curvature" R_#_#_1_2 [| [| 0 (* a '(+ (* a (cos θ)) b) (cos θ)) |] [| (* -1 '(+ (* a (cos θ)) b) a (cos θ)) 0 |] |]_#_#)
(assert-equal "Riemann curvature" R_#_#_2_1 [| [| 0 (* -1 a '(+ (* a (cos θ)) b) (cos θ)) |] [| (* '(+ (* a (cos θ)) b) a (cos θ)) 0 |] |]_#_#)
(assert-equal "Riemann curvature" R_#_#_2_2 [| [| 0 0 |] [| 0 0 |] |]_#_#)

;;
;; Ricci curvature
;;

(define $Ric__ (with-symbols {i} (contract + R~i_#_i_#)))

(assert-equal "Ricci curvature" Ric_#_# [| [| (/ (* a (cos θ)) '(+ (* a (cos θ)) b)) 0 |] [| 0 (/ (* '(+ (* a (cos θ)) b) (cos θ)) a) |] |]_#_#)

;;
;; Scalar curvature
;;

(define $scalar-curvature (with-symbols {j k} (. g~j~k Ric_j_k)))

(assert-equal "Scalar curvature" scalar-curvature (/ (* 2 (cos θ)) (* a '(+ (* a (cos θ)) b))))

;;
;; Covariant derivative of Riemann curvature tensor
;;

(define $∇R_____
  (with-symbols {i j k l m n}
    (- (∂/∂ R_i_j_k_l x~m)
       (. Γ~n_m_i R_n_j_k_l)
       (. Γ~n_m_j R_i_n_k_l)
       (. Γ~n_m_k R_i_j_n_l)
       (. Γ~n_m_l R_i_j_k_n))))

(assert-equal "Covariant derivative of Riemann curvature tensor"
  ∇R_#_#_#_#_#
  (tensor {2 2 2 2 2} {0 0 0 0 0 0 0 0 0 0 (+ (* -1 a '(+ (* a (cos θ)) b) (sin θ)) (* a^2 (sin θ) (cos θ))) 0 (+ (* a '(+ (* a (cos θ)) b) (sin θ)) (* -1 a^2 (sin θ) (cos θ))) 0 0 0 0 0 (+ (* '(+ (* a (cos θ)) b) a (sin θ)) (* -1 a^2 (sin θ) (cos θ))) 0 (+ (* -1 '(+ (* a (cos θ)) b) a (sin θ)) (* a^2 (sin θ) (cos θ))) 0 0 0 0 0 0 0 0 0 0 0} )_#_#_#_#_#)

;;
;; Second Bianchi identity
;;

(assert-equal "Second Bianchi identity"
  (with-symbols {i j k l m} (+ ∇R_i_j_k_l_m ∇R_i_j_l_m_k ∇R_i_j_m_k_l))
  (tensor {2 2 2 2 2} {0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0} ))