与えられた曲面に対して， 中心アファイン誘導接続と中心アファイン計量を計算します．

pu[f_][u_, v_] := D[f[uu, v], uu] /. uu -> u
pv[f_][u_, v_] := D[f[u, vv], vv] /. vv -> v
puu[f_][u_, v_] := D[f[uu, vv], {uu, 2}] /. {uu -> u, vv -> v}
puv[f_][u_, v_] := D[D[f[uu, vv], uu] /. uu -> u, vv] /. vv -> v
pvv[f_][u_, v_] := D[f[uu, vv], {vv, 2}] /. {uu -> u, vv -> v}
caframe[f_][u_, v_] := Transpose[{f[u, v], pu[f][u, v], pv[f][u, v]}]
m[f_][u_, v_] := Simplify[Inverse[caframe[f][u, v]].Transpose[{puu[f][u, v], puv[f][u, v], pvv[f][u, v]}] ]
cametric[f_][u_, v_] := {{m[f][u, v][[1, 1]], m[f][u, v][[1, 2]]}, {m[f][u, v][[1, 2]], m[f][u, v][[1, 3]]}}
caconnection11[f_][u_, v_] := {m[f][u, v][[2, 1]], m[f][u, v][[3, 1]]}
caconnection12[f_][u_, v_] := {m[f][u, v][[2, 2]], m[f][u, v][[3, 2]]}
caconnection22[f_][u_, v_] := {m[f][u, v][[2, 3]], m[f][u, v][[3, 3]]}

つぎの例に対して，計算してみましょう．

wang[a_, b_][u_, v_] := {u, v, u^(-a) v^(-b)}
parab[u_, v_] := {u, v, (1/2)( u^2 + v^2) - 1}

MatrixForm[Simplify[cametric[wang[1, 1]][u, v]]]
caconnection11[wang[1, 1]][u, v]
caconnection12[wang[1, 1]][u, v]
caconnection22[wang[1, 1]][u, v]
MatrixForm[Simplify[cametric[parab][u, v]]]
caconnection11[parab][u, v]
caconnection12[parab][u, v]
caconnection22[parab][u, v]

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