update

sample/math/geometry/riemann-curvature-tensor-of-M5-conformal.egi

@@ -0,0 +1,76 @@
+;;;
+;;; Polar coordinates
+;;;
+
+(define $x [|α β γ δ ε|])
+
+;;
+;; Metric tensor
+;;
+
+(define $g__ (generate-tensor 2#(* (a α β γ δ ε) G_%1_%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~___
+  (+ (- (∂/∂ Γ~i_j_k x_l) (∂/∂ Γ~i_j_l x_k))
+     (- (. Γ~m_j_k Γ~i_m_l) (. Γ~m_j_l Γ~i_m_k))))
+
+R~#_#_#_#
+
+;;
+;; Ricci curvature
+;;
+
+(define $Ric__ (with-symbols {i j k} (contract + R~i_j_k_i)))
+
+Ric_#_#
+
+;;
+;; Scalar curvature
+;;
+
+(define $scalar-curvature (with-symbols {j k} (. g~j~k Ric_j_k)))
+
+scalar-curvature
+
+;;
+;; Wodzicki-Chern-Simons form
+;;
+
+(let* {[[$es $os] (even-and-odd-permutations 5)]
+       [$es' (map 1#(lambda [$i] (nth i %1)) es)]
+       [$os' (map 1#(lambda [$i] (nth i %1)) os)]}
+  (- (sum (map (lambda [$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'))))