;; ;; Parameters ;; (define \$x [| φ θ ψ y α |]) ;; ;; Riemann metric of S2 x S3 ;; (define \$g__ (* (a φ θ ψ y α)^2 [|[| (/ (+ (* 3 '(+ 1 (* -1 y))^2 (sin θ)^2 '(+ a (* -1 y^2))) (* 2 '(+ a (* -3 y^2) (* 2 y^3)) (cos θ)^2 '(+ 1 (* -1 y))) (* '(+ a (* -2 y) y^2)^2 (cos θ)^2)) (* 18 '(+ a (* -1 y^2)) '(+ 1 (* -1 y)))) 0 (/ (+ (* -2 '(+ a (* -3 y^2) (* 2 y^3)) (cos θ) '(+ 1 (* -1 y))) (* -1 '(+ a (* -2 y) y^2)^2 (cos θ))) (* 18 '(+ a (* -1 y^2)) '(+ 1 (* -1 y)))) 0 (/ (* -1 '(+ a (* -2 y) y^2) (cos θ)) (* 3 '(+ 1 (* -1 y)))) |] [| 0 (/ '(+ 1 (* -1 y)) 6) 0 0 0 |] [| (/ (+ (* -2 '(+ a (* -3 y^2) (* 2 y^3)) (cos θ) '(+ 1 (* -1 y))) (* -1 '(+ a (* -2 y) y^2)^2 (cos θ))) (* 18 '(+ a (* -1 y^2)) '(+ 1 (* -1 y)))) 0 (/ (+ (* 2 '(+ a (* -3 y^2) (* 2 y^3)) '(+ 1 (* -1 y))) '(+ a (* -2 y) y^2)^2) (* 18 '(+ a (* -1 y^2)) '(+ 1 (* -1 y)))) 0 (/ (* 1 '(+ a (* -2 y) y^2)) (* 3 '(+ 1 (* -1 y)))) |] [| 0 0 0 (/ '(+ 1 (* -1 y)) (* 2 '(+ a (* -3 y^2) (* 2 y^3)))) 0 |] [| (/ (* -1 '(+ a (* -2 y) y^2) (cos θ)) (* 3 '(+ 1 (* -1 y)))) 0 (/ (* 1 '(+ a (* -2 y) y^2)) (* 3 '(+ 1 (* -1 y)))) 0 (/ (* 2 '(+ a (* -1 y^2))) '(+ 1 (* -1 y))) |] |]_#_#)) (define \$g~~ (M.inverse g_#_#)) g~#~# ;; ;; 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~#_#_#_# ;; ;; Ricci curvature ;; (define \$Ric__ (with-symbols {i} (contract + R~i_#_i_#))) Ric_#_# ;; ;; Wodzicki-Chern-Simons class ;; (define \$ret (let {[[\$es \$os] (even-and-odd-permutations 5)]} (- (sum' (map (lambda [\$σ] (debug (.' R~u_5_s_(σ 1) R~s_t_(σ 3)_(σ 2) R~t_u_(σ 5)_(σ 4)))) es)) (sum' (map (lambda [\$σ] (debug (.' R~u_5_s_(σ 1) R~s_t_(σ 3)_(σ 2) R~t_u_(σ 5)_(σ 4)))) os))))) ret (define \$ret2 (/ (expand-all' (numerator ret)) (denominator ret))) ret2 (define \$ret3 (/ (2#%1 (P./ (numerator ret2) (* (+ 1 (* -1 y))^3 (+ a (* -1 y^2))^5) y)) (/ (denominator ret2) (* '(+ 1 (* -1 y))^3 '(+ a (* -1 y^2))^5)))) ret3