riemann-curvature-tensor-of-S5-weyl.egi

;;;
;;; Parameters
;;;

(define $x [|θ φ ψ η ζ|])

(define $X [|(* r (cos θ))
             (* r (sin θ) (cos φ))
             (* r (sin θ) (sin φ) (cos ψ))
             (* r (sin θ) (sin φ) (sin ψ) (cos η))
             (* r (sin θ) (sin φ) (sin ψ) (sin η) (cos ζ))
             (* r (sin θ) (sin φ) (sin ψ) (sin η) (sin ζ))
             |])

;;
;; Local basis
;;

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

;;
;; Metric tensor
;;

(define $g__ (generate-tensor 2#(V.* e_%1 e_%2) {5 5}))
(define $g~~ (M.inverse g_#_#))
g_#_#
g~#~#

(with-symbols {i j k} (. g~i~j g_j_k))
;[| [| 1 0 0 0 0 |] [| 0 1 0 0 0 |] [| 0 0 1 0 0 |] [| 0 0 0 1 0 |] [| 0 0 0 0 1 |] |]

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

(define $Γ___
  (with-symbols {j k l}
    (* (/ 1 2)
       (+ (∂/∂ g_j_l x~k)
          (∂/∂ g_j_k x~l)
          (* -1 (∂/∂ g_k_l x~j))))))

Γ_#_#_#

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

(define $Γ~__
  (with-symbols {i j k l}
    (. g~i~j Γ_j_k_l)))

Γ~#_#_#

;;
;; 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~#_#_#_#
;(tensor {5 5 5 5} {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 (* -1 (sin θ)^2) 0 0 0 (sin θ)^2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -1 (sin θ)^2 (sin φ)^2) 0 0 0 0 0 0 0 (* (sin θ)^2 (sin φ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -1 (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 0 0 0 0 0 0 0 0 0 0 (* (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -1 (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2) 0 0 0 0 0 1 0 0 0 -1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -1 (sin θ)^2 (sin φ)^2) 0 0 0 (* (sin θ)^2 (sin φ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -1 (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 0 0 0 0 0 0 (* (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -1 (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2) 0 0 0 0 0 0 0 0 0 0 0 (* (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2) 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (sin θ)^2 0 0 0 (* -1 (sin θ)^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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -1 (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 0 0 (* (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -1 (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2) 0 0 0 0 0 0 0 (* (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2) 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (sin θ)^2 0 0 0 0 0 0 0 (* -1 (sin θ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* (sin θ)^2 (sin φ)^2) 0 0 0 (* -1 (sin θ)^2 (sin φ)^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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -1 (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2) 0 0 0 (* (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2) 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 (sin θ)^2 0 0 0 0 0 0 0 0 0 0 0 (* -1 (sin θ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* (sin θ)^2 (sin φ)^2) 0 0 0 0 0 0 0 (* -1 (sin θ)^2 (sin φ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 0 0 (* -1 (sin θ)^2 (sin φ)^2 (sin ψ)^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} )~#_#_#_#

;;
;; Ricci curvature
;;

(define $Ric__ (with-symbols {i j k} (contract + R~i_j_k_i)))

Ric_#_#

;;
;; Scalar curvature
;;

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

scalar-curvature

;;
;; Weyl curvature tensor
;;
(define $δ [| [| 1 0 0 0 0 |] [| 0 1 0 0 0 |] [| 0 0 1 0 0 |] [| 0 0 0 1 0 |] [| 0 0 0 0 1 |] |])
(define $Ric~_ (with-symbols {i k h} (. g~i~h Ric_k_h)))

(define $C~___
  (with-symbols {i j k l}
    (+ R~i_j_k_l
       (* (/ -1 3) (+ (- (. δ~i_k Ric_j_l) (. δ~i_l Ric_j_k))
                      (- (. Ric~i_k g_j_l) (. Ric~i_l g_j_k))))
       (* (/ scalar-curvature 12) (- (. δ~i_k g_j_l) (. δ~i_l g_j_k))))))

C~#_#_#_#
;(tensor {5 5 5 5} {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 (* -2 (sin θ)^2) 0 0 0 (* 2 (sin θ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -2 (sin θ)^2 (sin φ)^2) 0 0 0 0 0 0 0 (* 2 (sin θ)^2 (sin φ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -2 (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 0 0 0 0 0 0 0 0 0 0 (* 2 (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -2 (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* 2 (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2) 0 0 0 0 0 2 0 0 0 -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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -2 (sin θ)^2 (sin φ)^2) 0 0 0 (* 2 (sin θ)^2 (sin φ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -2 (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 0 0 0 0 0 0 (* 2 (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -2 (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2) 0 0 0 0 0 0 0 0 0 0 0 (* 2 (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2) 0 0 0 0 0 2 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* 2 (sin θ)^2) 0 0 0 (* -2 (sin θ)^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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -2 (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 0 0 (* 2 (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -2 (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2) 0 0 0 0 0 0 0 (* 2 (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2) 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* 2 (sin θ)^2) 0 0 0 0 0 0 0 (* -2 (sin θ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* 2 (sin θ)^2 (sin φ)^2) 0 0 0 (* -2 (sin θ)^2 (sin φ)^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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -2 (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2) 0 0 0 (* 2 (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2) 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 (* 2 (sin θ)^2) 0 0 0 0 0 0 0 0 0 0 0 (* -2 (sin θ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* 2 (sin θ)^2 (sin φ)^2) 0 0 0 0 0 0 0 (* -2 (sin θ)^2 (sin φ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* 2 (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 0 0 (* -2 (sin θ)^2 (sin φ)^2 (sin ψ)^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} )~#_#_#_#

;;
;; Wodzicki-Chern-Simons class
;;

(let {[[$es $os] (even-and-odd-permutations 5)]}
  (- (sum' (map (lambda [$σ] (.' C~u_1_s_(σ 1) C~s_t_(σ 3)_(σ 2) C~t_u_(σ 5)_(σ 4))) es))
     (sum' (map (lambda [$σ] (.' C~u_1_s_(σ 1) C~s_t_(σ 3)_(σ 2) C~t_u_(σ 5)_(σ 4))) os))))
;0