riemann-curvature-tensor-of-S5.egi

;;;
;;; Parameters
;;;

(define $x [|θ φ ψ η δ|])

(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 δ))
             (* r (sin θ) (sin φ) (sin ψ) (sin η) (sin δ))
             |])

;;
;; 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 δ)) (* r (cos θ) (sin φ) (sin ψ) (sin η) (sin δ)) |]
;  [| 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 δ)) (* r (sin θ) (cos φ) (sin ψ) (sin η) (sin δ)) |]
;  [| 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 δ)) (* r (sin θ) (sin φ) (cos ψ) (sin η) (sin δ)) |]
;  [| 0 0 0 (* -1 r (sin θ) (sin φ) (sin ψ) (sin η)) (* r (sin θ) (sin φ) (sin ψ) (cos η) (cos δ)) (* r (sin θ) (sin φ) (sin ψ) (cos η) (sin δ)) |]
;  [| 0 0 0 0 (* -1 r (sin θ) (sin φ) (sin ψ) (sin η) (sin δ)) (* r (sin θ) (sin φ) (sin ψ) (sin η) (cos δ)) |] |]

;;
;; Metric tensor
;;

(define $g__ (generate-tensor 2#(V.* e_%1 e_%2) {5 5}))
(define $g~~ (M.inverse g_#_#))
g_#_#;[| [| r^2 0 0 0 0 |] [| 0 (* r^2 (sin θ)^2) 0 0 0 |] [| 0 0 (* r^2 (sin θ)^2 (sin φ)^2) 0 0 |] [| 0 0 0 (* r^2 (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 |] [| 0 0 0 0 (* r^2 (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2) |] |]_#_#
g~#~#;[| [| (/ 1 r^2) 0 0 0 0 |] [| 0 (/ 1 (* r^2 (sin θ)^2)) 0 0 0 |] [| 0 0 (/ 1 (* r^2 (sin θ)^2 (sin φ)^2)) 0 0 |] [| 0 0 0 (/ 1 (* r^2 (sin θ)^2 (sin φ)^2 (sin ψ)^2)) 0 |] [| 0 0 0 0 (/ 1 (* r^2 (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2)) |] |]~#~#

(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_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} (. g~#~i Γ_i_#_#)))

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~#_#_#_#

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

R_#_#_#_#

;;
;; Ricci curvature
;;

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

Ric_#_#;[| [| 4 0 0 0 0 |] [| 0 (* 4 (sin θ)^2) 0 0 0 |] [| 0 0 (* 4 (sin θ)^2 (sin φ)^2) 0 0 |] [| 0 0 0 (* 4 (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 |] [| 0 0 0 0 (* 4 (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2) |] |]_#_#

;;
;; Scalar curvature
;;

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

scalar-curvature;(/ 20 r^2)

;;
;; Conformal curvature tensor
;;

(define $C_i_k_l_m
  (+ (. R_i_k_l_m)
     (+ (- (. Ric_i_m g_k_l) (. Ric_i_l g_k_m))
        (- (. Ric_k_l g_i_m) (. Ric_k_m g_i_l)))
     (* (/ scalar-curvature 2) (- (. g_i_l g_k_m) (. g_i_m g_k_l)))))

C_#_#_#_#

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

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