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RiemannTensor.wl
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(* ::Package:: *)
RiemannTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_]] :=
RiemannTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, index1, index2], False, True, True,
True] /; SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
RiemannTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, index1_, index2_], newCoordinates_List] :=
RiemannTensor[ResourceFunction["MetricTensor"][matrixRepresentation /. Thread[coordinates -> newCoordinates],
newCoordinates, index1, index2], False, True, True, True] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
Length[newCoordinates] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
RiemannTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
newCoordinates_List, index1_, index2_, index3_, index4_] :=
RiemannTensor[ResourceFunction["MetricTensor"][matrixRepresentation /. Thread[coordinates -> newCoordinates],
newCoordinates, metricIndex1, metricIndex2], index1, index2, index3, index4] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && Length[newCoordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] &&
BooleanQ[index4]
RiemannTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_, index3_, index4_]["TensorRepresentation"] :=
Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
christoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]];
riemannTensor = Normal[SparseArray[(Module[{index = #1}, index -> D[christoffelSymbols[[index[[1]],index[[2]],
index[[4]]]], newCoordinates[[index[[3]]]]] - D[christoffelSymbols[[index[[1]],index[[2]],index[[3]]]],
newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,index[[3]]]]*christoffelSymbols[[
#1,index[[2]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]] -
Total[(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ];
If[index1 === True && index2 === True && index3 === True && index4 === True,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,
index[[2]],index[[3]],index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]], If[index1 === False && index2 === False && index3 === False &&
index4 === False, Normal[SparseArray[
(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[2]],#1[[1]]]]*
Inverse[matrixRepresentation][[#1[[2]],index[[3]]]]*Inverse[matrixRepresentation][[#1[[3]],index[[4]]]]*
riemannTensor[[index[[1]],#1[[1]],#1[[2]],#1[[3]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
3]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]],
If[index1 === True && index2 === False && index3 === False && index4 === False,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*Inverse[matrixRepresentation][[#1[[3]],index[[3]]]]*
Inverse[matrixRepresentation][[#1[[4]],index[[4]]]]*riemannTensor[[#1[[1]],#1[[2]],#1[[3]],
#1[[4]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]], If[index1 === False && index2 === True &&
index3 === False && index4 === False, Normal[SparseArray[
(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[#1[[1]],index[[3]]]]*
Inverse[matrixRepresentation][[#1[[2]],index[[4]]]]*riemannTensor[[index[[1]],index[[2]],#1[[1]],
#1[[2]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]], If[index1 === False && index2 === False &&
index3 === True && index4 === False, Normal[SparseArray[(Module[{index = #1}, index ->
Total[(Inverse[matrixRepresentation][[index[[2]],#1[[1]]]]*Inverse[matrixRepresentation][[#1[[2]],
index[[4]]]]*riemannTensor[[index[[1]],#1[[1]],index[[3]],#1[[2]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
4]]], If[index1 === False && index2 === False && index3 === False && index4 === True,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[2]],#1[[1]]]]*
Inverse[matrixRepresentation][[#1[[2]],index[[3]]]]*riemannTensor[[index[[1]],#1[[1]],#1[[2]],
index[[4]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]], If[index1 === True && index2 === True &&
index3 === False && index4 === False, Normal[SparseArray[(Module[{index = #1}, index ->
Total[(matrixRepresentation[[index[[1]],#1[[1]]]]*Inverse[matrixRepresentation][[#1[[2]],index[[3]]]]*
Inverse[matrixRepresentation][[#1[[3]],index[[4]]]]*riemannTensor[[#1[[1]],index[[2]],#1[[2]],
#1[[3]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 3]]] & ) /@ Tuples[
Range[Length[matrixRepresentation]], 4]]], If[index1 === True && index2 === False && index3 === True &&
index4 === False, Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],
#1[[1]]]]*Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*Inverse[matrixRepresentation][[
#1[[3]],index[[4]]]]*riemannTensor[[#1[[1]],#1[[2]],index[[3]],#1[[3]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 3]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
4]]], If[index1 === True && index2 === False && index3 === False && index4 === True,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*Inverse[matrixRepresentation][[#1[[3]],
index[[3]]]]*riemannTensor[[#1[[1]],#1[[2]],#1[[3]],index[[4]]]] & ) /@ Tuples[
Range[Length[matrixRepresentation]], 3]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]],
If[index1 === False && index2 === True && index3 === True && index4 === False, Normal[
SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[#1,index[[4]]]]*
riemannTensor[[index[[1]],index[[2]],index[[3]],#1]] & ) /@ Range[Length[
matrixRepresentation]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]], If[
index1 === False && index2 === True && index3 === False && index4 === True,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[#1,index[[3]]]]*
riemannTensor[[index[[1]],index[[2]],#1,index[[4]]]] & ) /@ Range[Length[
matrixRepresentation]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]],
If[index1 === False && index2 === False && index3 === True && index4 === True,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[2]],#1]]*
riemannTensor[[index[[1]],#1,index[[3]],index[[4]]]] & ) /@ Range[Length[
matrixRepresentation]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]],
If[index1 === True && index2 === True && index3 === True && index4 === False, Normal[SparseArray[
(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[#1[[2]],index[[4]]]]*riemannTensor[[#1[[1]],index[[2]],
index[[3]],#1[[2]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]], If[index1 === True && index2 === True &&
index3 === False && index4 === True, Normal[SparseArray[(Module[{index = #1}, index ->
Total[(matrixRepresentation[[index[[1]],#1[[1]]]]*Inverse[matrixRepresentation][[#1[[2]],
index[[3]]]]*riemannTensor[[#1[[1]],index[[2]],#1[[2]],index[[4]]]] & ) /@ Tuples[
Range[Length[matrixRepresentation]], 2]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
4]]], If[index1 === True && index2 === False && index3 === True && index4 === True,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*riemannTensor[[#1[[1]],#1[[2]],
index[[3]],index[[4]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]], If[index1 === False && index2 === True &&
index3 === True && index4 === True, riemannTensor, Indeterminate]]]]]]]]]]]]]]]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] && BooleanQ[index4]
RiemannTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_, index3_, index4_]["ReducedTensorRepresentation"] :=
Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
christoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]];
riemannTensor = Normal[SparseArray[(Module[{index = #1}, index -> D[christoffelSymbols[[index[[1]],index[[2]],
index[[4]]]], newCoordinates[[index[[3]]]]] - D[christoffelSymbols[[index[[1]],index[[2]],index[[3]]]],
newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,index[[3]]]]*christoffelSymbols[[
#1,index[[2]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]] -
Total[(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ];
If[index1 === True && index2 === True && index3 === True && index4 === True,
FullSimplify[Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],#1]]*
riemannTensor[[#1,index[[2]],index[[3]],index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]], If[index1 === False && index2 === False &&
index3 === False && index4 === False, FullSimplify[
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[2]],#1[[1]]]]*
Inverse[matrixRepresentation][[#1[[2]],index[[3]]]]*Inverse[matrixRepresentation][[#1[[3]],index[[4]]]]*
riemannTensor[[index[[1]],#1[[1]],#1[[2]],#1[[3]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
3]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]]],
If[index1 === True && index2 === False && index3 === False && index4 === False,
FullSimplify[Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*Inverse[matrixRepresentation][[#1[[3]],
index[[3]]]]*Inverse[matrixRepresentation][[#1[[4]],index[[4]]]]*riemannTensor[[#1[[1]],#1[[2]],
#1[[3]],#1[[4]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]], If[index1 === False && index2 === True &&
index3 === False && index4 === False, FullSimplify[
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[#1[[1]],index[[3]]]]*
Inverse[matrixRepresentation][[#1[[2]],index[[4]]]]*riemannTensor[[index[[1]],index[[2]],#1[[1]],
#1[[2]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]], If[index1 === False && index2 === False &&
index3 === True && index4 === False, FullSimplify[Normal[SparseArray[
(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[2]],#1[[1]]]]*
Inverse[matrixRepresentation][[#1[[2]],index[[4]]]]*riemannTensor[[index[[1]],#1[[1]],index[[3]],
#1[[2]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]], If[index1 === False && index2 === False &&
index3 === False && index4 === True, FullSimplify[Normal[SparseArray[(Module[{index = #1},
index -> Total[(Inverse[matrixRepresentation][[index[[2]],#1[[1]]]]*Inverse[matrixRepresentation][[
#1[[2]],index[[3]]]]*riemannTensor[[index[[1]],#1[[1]],#1[[2]],index[[4]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
4]]]], If[index1 === True && index2 === True && index3 === False && index4 === False,
FullSimplify[Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],
#1[[1]]]]*Inverse[matrixRepresentation][[#1[[2]],index[[3]]]]*Inverse[matrixRepresentation][[
#1[[3]],index[[4]]]]*riemannTensor[[#1[[1]],index[[2]],#1[[2]],#1[[3]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 3]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
4]]]], If[index1 === True && index2 === False && index3 === True && index4 === False,
FullSimplify[Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],
#1[[1]]]]*Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*Inverse[matrixRepresentation][[
#1[[3]],index[[4]]]]*riemannTensor[[#1[[1]],#1[[2]],index[[3]],#1[[3]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 3]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
4]]]], If[index1 === True && index2 === False && index3 === False && index4 === True,
FullSimplify[Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],
#1[[1]]]]*Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*Inverse[matrixRepresentation][[
#1[[3]],index[[3]]]]*riemannTensor[[#1[[1]],#1[[2]],#1[[3]],index[[4]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 3]]] & ) /@ Tuples[Range[Length[
matrixRepresentation]], 4]]]], If[index1 === False && index2 === True && index3 === True &&
index4 === False, FullSimplify[Normal[SparseArray[(Module[{index = #1}, index -> Total[
(Inverse[matrixRepresentation][[#1,index[[4]]]]*riemannTensor[[index[[1]],index[[2]],index[[3]],
#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@ Tuples[Range[Length[
matrixRepresentation]], 4]]]], If[index1 === False && index2 === True && index3 === False &&
index4 === True, FullSimplify[Normal[SparseArray[(Module[{index = #1}, index -> Total[
(Inverse[matrixRepresentation][[#1,index[[3]]]]*riemannTensor[[index[[1]],index[[3]],#1,
index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@ Tuples[
Range[Length[matrixRepresentation]], 4]]]], If[index1 === False && index2 === False &&
index3 === True && index4 === True, FullSimplify[Normal[SparseArray[(Module[{index = #1},
index -> Total[(Inverse[matrixRepresentation][[index[[2]],#1]]*riemannTensor[[index[[1]],#1,
index[[3]],index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]], If[index1 === True && index2 === True &&
index3 === True && index4 === False, FullSimplify[Normal[SparseArray[(Module[{index = #1},
index -> Total[(matrixRepresentation[[index[[1]],#1[[1]]]]*Inverse[matrixRepresentation][[#1[[2]],
index[[4]]]]*riemannTensor[[#1[[1]],index[[2]],index[[3]],#1[[2]]]] & ) /@ Tuples[
Range[Length[matrixRepresentation]], 2]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
4]]]], If[index1 === True && index2 === True && index3 === False && index4 === True,
FullSimplify[Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],
#1[[1]]]]*Inverse[matrixRepresentation][[#1[[2]],index[[3]]]]*riemannTensor[[#1[[1]],
index[[2]],#1[[2]],index[[4]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
2]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]]], If[index1 === True &&
index2 === False && index3 === True && index4 === True, FullSimplify[Normal[SparseArray[
(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*riemannTensor[[#1[[1]],#1[[2]],
index[[3]],index[[4]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]], If[index1 === False && index2 === True &&
index3 === True && index4 === True, FullSimplify[riemannTensor], Indeterminate]]]]]]]]]]]]]]]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] && BooleanQ[index4]
RiemannTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_, index3_, index4_]["SymbolicTensorRepresentation"] :=
Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
christoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(Inactive[D][newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] +
Inactive[D][newMatrixRepresentation[[index[[2]],#1]], newCoordinates[[index[[3]]]]] -
Inactive[D][newMatrixRepresentation[[index[[2]],index[[3]]]], newCoordinates[[#1]]]) & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 3]]];
riemannTensor = Normal[SparseArray[(Module[{index = #1}, index -> Inactive[D][christoffelSymbols[[index[[1]],
index[[2]],index[[4]]]], newCoordinates[[index[[3]]]]] - Inactive[D][christoffelSymbols[[index[[1]],
index[[2]],index[[3]]]], newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,
index[[3]]]]*christoffelSymbols[[#1,index[[2]],index[[4]]]] & ) /@
Range[Length[newMatrixRepresentation]]] - Total[(christoffelSymbols[[index[[1]],#1,index[[4]]]]*
christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; If[index1 === True && index2 === True && index3 === True && index4 === True,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,
index[[2]],index[[3]],index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]], If[index1 === False && index2 === False && index3 === False &&
index4 === False, Normal[SparseArray[
(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[2]],#1[[1]]]]*
Inverse[matrixRepresentation][[#1[[2]],index[[3]]]]*Inverse[matrixRepresentation][[#1[[3]],index[[4]]]]*
riemannTensor[[index[[1]],#1[[1]],#1[[2]],#1[[3]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
3]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]],
If[index1 === True && index2 === False && index3 === False && index4 === False,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*Inverse[matrixRepresentation][[#1[[3]],index[[3]]]]*
Inverse[matrixRepresentation][[#1[[4]],index[[4]]]]*riemannTensor[[#1[[1]],#1[[2]],#1[[3]],
#1[[4]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]], If[index1 === False && index2 === True &&
index3 === False && index4 === False, Normal[SparseArray[
(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[#1[[1]],index[[3]]]]*
Inverse[matrixRepresentation][[#1[[2]],index[[4]]]]*riemannTensor[[index[[1]],index[[2]],#1[[1]],
#1[[2]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]], If[index1 === False && index2 === False &&
index3 === True && index4 === False, Normal[SparseArray[(Module[{index = #1}, index ->
Total[(Inverse[matrixRepresentation][[index[[2]],#1[[1]]]]*Inverse[matrixRepresentation][[#1[[2]],
index[[4]]]]*riemannTensor[[index[[1]],#1[[1]],index[[3]],#1[[2]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
4]]], If[index1 === False && index2 === False && index3 === False && index4 === True,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[2]],#1[[1]]]]*
Inverse[matrixRepresentation][[#1[[2]],index[[3]]]]*riemannTensor[[index[[1]],#1[[1]],#1[[2]],
index[[4]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]], If[index1 === True && index2 === True &&
index3 === False && index4 === False, Normal[SparseArray[(Module[{index = #1}, index ->
Total[(matrixRepresentation[[index[[1]],#1[[1]]]]*Inverse[matrixRepresentation][[#1[[2]],index[[3]]]]*
Inverse[matrixRepresentation][[#1[[3]],index[[4]]]]*riemannTensor[[#1[[1]],index[[2]],#1[[2]],
#1[[3]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 3]]] & ) /@ Tuples[
Range[Length[matrixRepresentation]], 4]]], If[index1 === True && index2 === False && index3 === True &&
index4 === False, Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],
#1[[1]]]]*Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*Inverse[matrixRepresentation][[
#1[[3]],index[[4]]]]*riemannTensor[[#1[[1]],#1[[2]],index[[3]],#1[[3]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 3]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
4]]], If[index1 === True && index2 === False && index3 === False && index4 === True,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*Inverse[matrixRepresentation][[#1[[3]],
index[[3]]]]*riemannTensor[[#1[[1]],#1[[2]],#1[[3]],index[[4]]]] & ) /@ Tuples[
Range[Length[matrixRepresentation]], 3]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]],
If[index1 === False && index2 === True && index3 === True && index4 === False, Normal[
SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[#1,index[[4]]]]*
riemannTensor[[index[[1]],index[[2]],index[[3]],#1]] & ) /@ Range[Length[
matrixRepresentation]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]], If[
index1 === False && index2 === True && index3 === False && index4 === True,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[#1,index[[3]]]]*
riemannTensor[[index[[1]],index[[2]],#1,index[[4]]]] & ) /@ Range[Length[
matrixRepresentation]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]],
If[index1 === False && index2 === False && index3 === True && index4 === True,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[2]],#1]]*
riemannTensor[[index[[1]],#1,index[[3]],index[[4]]]] & ) /@ Range[Length[
matrixRepresentation]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]],
If[index1 === True && index2 === True && index3 === True && index4 === False, Normal[SparseArray[
(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[#1[[2]],index[[4]]]]*riemannTensor[[#1[[1]],index[[2]],
index[[3]],#1[[2]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]], If[index1 === True && index2 === True &&
index3 === False && index4 === True, Normal[SparseArray[(Module[{index = #1}, index ->
Total[(matrixRepresentation[[index[[1]],#1[[1]]]]*Inverse[matrixRepresentation][[#1[[2]],
index[[3]]]]*riemannTensor[[#1[[1]],index[[2]],#1[[2]],index[[4]]]] & ) /@ Tuples[
Range[Length[matrixRepresentation]], 2]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
4]]], If[index1 === True && index2 === False && index3 === True && index4 === True,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*riemannTensor[[#1[[1]],#1[[2]],
index[[3]],index[[4]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]], If[index1 === False && index2 === True &&
index3 === True && index4 === True, riemannTensor, Indeterminate]]]]]]]]]]]]]]]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] && BooleanQ[index4]
RiemannTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_, index3_, index4_]["KretschmannScalar"] :=
Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, covariantRiemannTensor,
contravariantRiemannTensor}, newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; christoffelSymbols =
Normal[SparseArray[(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]];
riemannTensor = Normal[SparseArray[(Module[{index = #1}, index -> D[christoffelSymbols[[index[[1]],index[[2]],
index[[4]]]], newCoordinates[[index[[3]]]]] - D[christoffelSymbols[[index[[1]],index[[2]],index[[3]]]],
newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,index[[3]]]]*christoffelSymbols[[
#1,index[[2]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]] -
Total[(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ]; covariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,
index[[2]],index[[3]],index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]; contravariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[2]],#1[[1]]]]*
Inverse[matrixRepresentation][[#1[[2]],index[[3]]]]*Inverse[matrixRepresentation][[#1[[3]],index[[4]]]]*
riemannTensor[[index[[1]],#1[[1]],#1[[2]],#1[[3]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
3]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]];
Total[(covariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*contravariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],
#1[[4]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] && BooleanQ[index4]
RiemannTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_, index3_, index4_]["ReducedKretschmannScalar"] :=
Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, covariantRiemannTensor,
contravariantRiemannTensor}, newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; christoffelSymbols =
Normal[SparseArray[(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]];
riemannTensor = Normal[SparseArray[(Module[{index = #1}, index -> D[christoffelSymbols[[index[[1]],index[[2]],
index[[4]]]], newCoordinates[[index[[3]]]]] - D[christoffelSymbols[[index[[1]],index[[2]],index[[3]]]],
newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,index[[3]]]]*christoffelSymbols[[
#1,index[[2]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]] -
Total[(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ]; covariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,
index[[2]],index[[3]],index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]; contravariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[2]],#1[[1]]]]*
Inverse[matrixRepresentation][[#1[[2]],index[[3]]]]*Inverse[matrixRepresentation][[#1[[3]],index[[4]]]]*
riemannTensor[[index[[1]],#1[[1]],#1[[2]],#1[[3]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
3]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]];
FullSimplify[Total[(covariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*contravariantRiemannTensor[[#1[[1]],
#1[[2]],#1[[3]],#1[[4]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] && BooleanQ[index4]
RiemannTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_, index3_, index4_]["SymbolicKretschmannScalar"] :=
Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, covariantRiemannTensor,
contravariantRiemannTensor}, newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; christoffelSymbols =
Normal[SparseArray[(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(Inactive[D][newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] +
Inactive[D][newMatrixRepresentation[[index[[2]],#1]], newCoordinates[[index[[3]]]]] -
Inactive[D][newMatrixRepresentation[[index[[2]],index[[3]]]], newCoordinates[[#1]]]) & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 3]]];
riemannTensor = Normal[SparseArray[(Module[{index = #1}, index -> Inactive[D][christoffelSymbols[[index[[1]],
index[[2]],index[[4]]]], newCoordinates[[index[[3]]]]] - Inactive[D][christoffelSymbols[[index[[1]],
index[[2]],index[[3]]]], newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,
index[[3]]]]*christoffelSymbols[[#1,index[[2]],index[[4]]]] & ) /@
Range[Length[newMatrixRepresentation]]] - Total[(christoffelSymbols[[index[[1]],#1,index[[4]]]]*
christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; covariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,
index[[2]],index[[3]],index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]; contravariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[2]],#1[[1]]]]*
Inverse[matrixRepresentation][[#1[[2]],index[[3]]]]*Inverse[matrixRepresentation][[#1[[3]],index[[4]]]]*
riemannTensor[[index[[1]],#1[[1]],#1[[2]],#1[[3]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
3]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]];
Total[(covariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*contravariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],
#1[[4]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] && BooleanQ[index4]
RiemannTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_, index3_, index4_]["ChernPontryaginScalar"] :=
Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, covariantRiemannTensor,
mixedRiemannTensor, leviCivitaTensor, contravariantLeviCivitaTensor}, If[Length[matrixRepresentation] == 4,
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
christoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]]; riemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> D[christoffelSymbols[[index[[1]],index[[2]],index[[4]]]],
newCoordinates[[index[[3]]]]] - D[christoffelSymbols[[index[[1]],index[[2]],index[[3]]]],
newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,index[[3]]]]*christoffelSymbols[[
#1,index[[2]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]] - Total[
(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ]; covariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,
index[[2]],index[[3]],index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]; mixedRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[2]],#1]]*
riemannTensor[[index[[1]],#1,index[[3]],index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]; leviCivitaTensor = Normal[LeviCivitaTensor[4]];
contravariantLeviCivitaTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*Inverse[matrixRepresentation][[#1[[3]],index[[3]]]]*
Inverse[matrixRepresentation][[#1[[4]],index[[4]]]]*leviCivitaTensor[[#1[[1]],#1[[2]],#1[[3]],
#1[[4]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]];
Total[(covariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*contravariantLeviCivitaTensor[[#1[[1]],#1[[2]],
#1[[5]],#1[[6]]]]*mixedRiemannTensor[[#1[[3]],#1[[4]],#1[[5]],#1[[6]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 6]], Indeterminate]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] && BooleanQ[index4]
RiemannTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_, index3_, index4_]["ReducedChernPontryaginScalar"] :=
Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, covariantRiemannTensor,
mixedRiemannTensor, leviCivitaTensor, contravariantLeviCivitaTensor}, If[Length[matrixRepresentation] == 4,
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
christoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]]; riemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> D[christoffelSymbols[[index[[1]],index[[2]],index[[4]]]],
newCoordinates[[index[[3]]]]] - D[christoffelSymbols[[index[[1]],index[[2]],index[[3]]]],
newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,index[[3]]]]*christoffelSymbols[[
#1,index[[2]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]] - Total[
(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ]; covariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,
index[[2]],index[[3]],index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]; mixedRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[2]],#1]]*
riemannTensor[[index[[1]],#1,index[[3]],index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]; leviCivitaTensor = Normal[LeviCivitaTensor[4]];
contravariantLeviCivitaTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*Inverse[matrixRepresentation][[#1[[3]],index[[3]]]]*
Inverse[matrixRepresentation][[#1[[4]],index[[4]]]]*leviCivitaTensor[[#1[[1]],#1[[2]],#1[[3]],
#1[[4]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]];
FullSimplify[Total[(covariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*contravariantLeviCivitaTensor[[
#1[[1]],#1[[2]],#1[[5]],#1[[6]]]]*mixedRiemannTensor[[#1[[3]],#1[[4]],#1[[5]],#1[[6]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 6]]], Indeterminate]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] && BooleanQ[index4]
RiemannTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_, index3_, index4_]["SymbolicChernPontryaginScalar"] :=
Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, covariantRiemannTensor,
mixedRiemannTensor, leviCivitaTensor, contravariantLeviCivitaTensor}, If[Length[matrixRepresentation] == 4,
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
christoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(Inactive[D][newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] +
Inactive[D][newMatrixRepresentation[[index[[2]],#1]], newCoordinates[[index[[3]]]]] -
Inactive[D][newMatrixRepresentation[[index[[2]],index[[3]]]], newCoordinates[[#1]]]) & ) /@ Range[
Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 3]]];
riemannTensor = Normal[SparseArray[(Module[{index = #1}, index -> Inactive[D][christoffelSymbols[[index[[1]],
index[[2]],index[[4]]]], newCoordinates[[index[[3]]]]] - Inactive[D][christoffelSymbols[[index[[1]],
index[[2]],index[[3]]]], newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,
index[[3]]]]*christoffelSymbols[[#1,index[[2]],index[[4]]]] & ) /@ Range[Length[
newMatrixRepresentation]]] - Total[(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[
#1,index[[2]],index[[3]]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; covariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,
index[[2]],index[[3]],index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]; mixedRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[2]],#1]]*
riemannTensor[[index[[1]],#1,index[[3]],index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]; leviCivitaTensor = Normal[LeviCivitaTensor[4]];
contravariantLeviCivitaTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*Inverse[matrixRepresentation][[#1[[3]],index[[3]]]]*
Inverse[matrixRepresentation][[#1[[4]],index[[4]]]]*leviCivitaTensor[[#1[[1]],#1[[2]],#1[[3]],
#1[[4]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]];
Total[(covariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*contravariantLeviCivitaTensor[[#1[[1]],#1[[2]],
#1[[5]],#1[[6]]]]*mixedRiemannTensor[[#1[[3]],#1[[4]],#1[[5]],#1[[6]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 6]], Indeterminate]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] && BooleanQ[index4]
RiemannTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_, index3_, index4_]["EulerScalar"] :=
Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, covariantRiemannTensor,
leviCivitaTensor, contravariantLeviCivitaTensor}, If[Length[matrixRepresentation] == 4,
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
christoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]]; riemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> D[christoffelSymbols[[index[[1]],index[[2]],index[[4]]]],
newCoordinates[[index[[3]]]]] - D[christoffelSymbols[[index[[1]],index[[2]],index[[3]]]],
newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,index[[3]]]]*christoffelSymbols[[
#1,index[[2]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]] - Total[
(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ]; covariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,
index[[2]],index[[3]],index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]; leviCivitaTensor = Normal[LeviCivitaTensor[4]];
contravariantLeviCivitaTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*Inverse[matrixRepresentation][[#1[[3]],index[[3]]]]*
Inverse[matrixRepresentation][[#1[[4]],index[[4]]]]*leviCivitaTensor[[#1[[1]],#1[[2]],#1[[3]],
#1[[4]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]];
Total[(covariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*covariantRiemannTensor[[#1[[5]],#1[[6]],#1[[7]],
#1[[8]]]]*contravariantLeviCivitaTensor[[#1[[1]],#1[[2]],#1[[5]],#1[[6]]]]*contravariantLeviCivitaTensor[[
#1[[3]],#1[[4]],#1[[7]],#1[[8]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 8]], Indeterminate]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] && BooleanQ[index4]
RiemannTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_, index3_, index4_]["ReducedEulerScalar"] :=
Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, covariantRiemannTensor,
leviCivitaTensor, contravariantLeviCivitaTensor}, If[Length[matrixRepresentation] == 4,
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
christoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]]; riemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> D[christoffelSymbols[[index[[1]],index[[2]],index[[4]]]],
newCoordinates[[index[[3]]]]] - D[christoffelSymbols[[index[[1]],index[[2]],index[[3]]]],
newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,index[[3]]]]*christoffelSymbols[[
#1,index[[2]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]] - Total[
(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ]; covariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,
index[[2]],index[[3]],index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]; leviCivitaTensor = Normal[LeviCivitaTensor[4]];
contravariantLeviCivitaTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*Inverse[matrixRepresentation][[#1[[3]],index[[3]]]]*
Inverse[matrixRepresentation][[#1[[4]],index[[4]]]]*leviCivitaTensor[[#1[[1]],#1[[2]],#1[[3]],
#1[[4]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]];
FullSimplify[Total[(covariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*covariantRiemannTensor[[#1[[5]],
#1[[6]],#1[[7]],#1[[8]]]]*contravariantLeviCivitaTensor[[#1[[1]],#1[[2]],#1[[5]],#1[[6]]]]*
contravariantLeviCivitaTensor[[#1[[3]],#1[[4]],#1[[7]],#1[[8]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 8]]], Indeterminate]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] && BooleanQ[index4]
RiemannTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_, index3_, index4_]["SymbolicEulerScalar"] :=
Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, covariantRiemannTensor,
leviCivitaTensor, contravariantLeviCivitaTensor}, If[Length[matrixRepresentation] == 4,
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
christoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(Inactive[D][newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] +
Inactive[D][newMatrixRepresentation[[index[[2]],#1]], newCoordinates[[index[[3]]]]] -
Inactive[D][newMatrixRepresentation[[index[[2]],index[[3]]]], newCoordinates[[#1]]]) & ) /@ Range[
Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 3]]];
riemannTensor = Normal[SparseArray[(Module[{index = #1}, index -> Inactive[D][christoffelSymbols[[index[[1]],
index[[2]],index[[4]]]], newCoordinates[[index[[3]]]]] - Inactive[D][christoffelSymbols[[index[[1]],
index[[2]],index[[3]]]], newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,
index[[3]]]]*christoffelSymbols[[#1,index[[2]],index[[4]]]] & ) /@ Range[Length[
newMatrixRepresentation]]] - Total[(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[
#1,index[[2]],index[[3]]]] & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 4]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; covariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,
index[[2]],index[[3]],index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]; leviCivitaTensor = Normal[LeviCivitaTensor[4]];
contravariantLeviCivitaTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*Inverse[matrixRepresentation][[#1[[3]],index[[3]]]]*
Inverse[matrixRepresentation][[#1[[4]],index[[4]]]]*leviCivitaTensor[[#1[[1]],#1[[2]],#1[[3]],
#1[[4]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]];
Total[(covariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*covariantRiemannTensor[[#1[[5]],#1[[6]],#1[[7]],
#1[[8]]]]*contravariantLeviCivitaTensor[[#1[[1]],#1[[2]],#1[[5]],#1[[6]]]]*contravariantLeviCivitaTensor[[
#1[[3]],#1[[4]],#1[[7]],#1[[8]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 8]], Indeterminate]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] && BooleanQ[index4]
RiemannTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_, index3_, index4_]["MetricTensor"] := ResourceFunction["MetricTensor"][matrixRepresentation, coordinates,
metricIndex1, metricIndex2] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] &&
BooleanQ[index4]
RiemannTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_, index3_, index4_]["Coordinates"] := coordinates /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] &&
BooleanQ[index4]
RiemannTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_, index3_, index4_]["CoordinateOneForms"] :=
(If[Head[#1] === Subscript, Subscript[StringJoin["\[FormalD]", ToString[First[#1]]], ToString[Last[#1]]],
If[Head[#1] === Superscript, Superscript[StringJoin["\[FormalD]", ToString[First[#1]]], ToString[Last[#1]]],
StringJoin["\[FormalD]", ToString[#1]]]] & ) /@ coordinates /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] &&
BooleanQ[index4]
RiemannTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_, index3_, index4_]["Indices"] := {index1, index2, index3, index4} /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] && BooleanQ[index4]
RiemannTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_, index3_, index4_]["CovariantQ"] :=
If[index1 === True && index2 === True && index3 === True && index4 === True, True, False] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] && BooleanQ[index4]
RiemannTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_, index3_, index4_]["ContravariantQ"] :=
If[index1 === False && index2 === False && index3 === False && index4 === False, True, False] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] && BooleanQ[index4]
RiemannTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_, index3_, index4_]["MixedQ"] :=
If[ !((index1 === True && index2 === True && index3 === True && index4 === True) ||
(index1 === False && index2 === False && index3 === False && index4 === False)), True, False] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] && BooleanQ[index4]
RiemannTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_, index3_, index4_]["Symbol"] := If[index1 === True && index2 === True && index3 === True && index4 === True,
Subscript["\[FormalCapitalR]", "\[FormalRho]\[FormalSigma]\[FormalMu]\[FormalNu]"], If[index1 === False && index2 === False && index3 === False && index4 === False,
Superscript["\[FormalCapitalR]", "\[FormalRho]\[FormalSigma]\[FormalMu]\[FormalNu]"], If[index1 === True && index2 === False && index3 === False && index4 === False,
Subsuperscript["\[FormalCapitalR]", "\[FormalRho]", "\[FormalSigma]\[FormalMu]\[FormalNu]"], If[index1 === False && index2 === True && index3 === False &&
index4 === False, Subsuperscript["\[FormalCapitalR]", "\[FormalSigma]", "\[FormalRho]\[FormalMu]\[FormalNu]"], If[index1 === False && index2 === False &&
index3 === True && index4 === False, Subsuperscript["\[FormalCapitalR]", "\[FormalMu]", "\[FormalRho]\[FormalSigma]\[FormalNu]"],
If[index1 === False && index2 === False && index3 === False && index4 === True,
Subsuperscript["\[FormalCapitalR]", "\[FormalNu]", "\[FormalRho]\[FormalSigma]\[FormalMu]"], If[index1 === True && index2 === True && index3 === False &&
index4 === False, Subsuperscript["\[FormalCapitalR]", "\[FormalRho]\[FormalSigma]", "\[FormalMu]\[FormalNu]"], If[index1 === True && index2 === False &&
index3 === True && index4 === False, Subsuperscript["\[FormalCapitalR]", "\[FormalRho]\[FormalMu]", "\[FormalSigma]\[FormalNu]"],
If[index1 === True && index2 === False && index3 === False && index4 === True, Subsuperscript["\[FormalCapitalR]", "\[FormalRho]\[FormalNu]",
"\[FormalSigma]\[FormalMu]"], If[index1 === False && index2 === True && index3 === True && index4 === False,
Subsuperscript["\[FormalCapitalR]", "\[FormalSigma]\[FormalMu]", "\[FormalRho]\[FormalNu]"], If[index1 === False && index2 === True && index3 === False &&
index4 === True, Subsuperscript["\[FormalCapitalR]", "\[FormalSigma]\[FormalNu]", "\[FormalRho]\[FormalMu]"], If[index1 === False && index2 === False &&
index3 === True && index4 === True, Subsuperscript["\[FormalCapitalR]", "\[FormalMu]\[FormalNu]", "\[FormalRho]\[FormalSigma]"], If[index1 === True &&
index2 === True && index3 === True && index4 === False, Subsuperscript["\[FormalCapitalR]", "\[FormalRho]\[FormalSigma]\[FormalMu]", "\[FormalNu]"],
If[index1 === True && index2 === True && index3 === False && index4 === True, Subsuperscript["\[FormalCapitalR]",
"\[FormalRho]\[FormalSigma]\[FormalNu]", "\[FormalMu]"], If[index1 === True && index2 === False && index3 === True && index4 === True,
Subsuperscript["\[FormalCapitalR]", "\[FormalRho]\[FormalMu]\[FormalNu]", "\[FormalSigma]"], If[index1 === False && index2 === True && index3 === True &&
index4 === True, Subsuperscript["\[FormalCapitalR]", "\[FormalSigma]\[FormalMu]\[FormalNu]", "\[FormalRho]"], Indeterminate]]]]]]]]]]]]]]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] && BooleanQ[index4]
RiemannTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_, index3_, index4_]["RiemannFlatQ"] :=
Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, fieldEquations},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
christoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]];
riemannTensor = Normal[SparseArray[(Module[{index = #1}, index -> D[christoffelSymbols[[index[[1]],index[[2]],
index[[4]]]], newCoordinates[[index[[3]]]]] - D[christoffelSymbols[[index[[1]],index[[2]],index[[3]]]],
newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,index[[3]]]]*christoffelSymbols[[
#1,index[[2]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]] -
Total[(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ];
fieldEquations = FullSimplify[Thread[Catenate[Catenate[Catenate[riemannTensor]]] ==
Catenate[Catenate[Catenate[ConstantArray[0, {Length[matrixRepresentation], Length[matrixRepresentation],
Length[matrixRepresentation], Length[matrixRepresentation]}]]]]]];
If[fieldEquations === True, True, If[fieldEquations === False, False,
If[Length[Select[fieldEquations, #1 === True & ]] == Length[matrixRepresentation]*Length[matrixRepresentation]*
Length[matrixRepresentation]*Length[matrixRepresentation], True, False]]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] && BooleanQ[index4]
RiemannTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_, index3_, index4_]["VanishingKretschmannScalarQ"] :=
Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, covariantRiemannTensor,
contravariantRiemannTensor}, newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; christoffelSymbols =
Normal[SparseArray[(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]];
riemannTensor = Normal[SparseArray[(Module[{index = #1}, index -> D[christoffelSymbols[[index[[1]],index[[2]],
index[[4]]]], newCoordinates[[index[[3]]]]] - D[christoffelSymbols[[index[[1]],index[[2]],index[[3]]]],
newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,index[[3]]]]*christoffelSymbols[[
#1,index[[2]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]] -
Total[(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ]; covariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,
index[[2]],index[[3]],index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]; contravariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[2]],#1[[1]]]]*
Inverse[matrixRepresentation][[#1[[2]],index[[3]]]]*Inverse[matrixRepresentation][[#1[[3]],index[[4]]]]*
riemannTensor[[index[[1]],#1[[1]],#1[[2]],#1[[3]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
3]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]];
FullSimplify[Total[(covariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*contravariantRiemannTensor[[#1[[1]],
#1[[2]],#1[[3]],#1[[4]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]] == 0] === True] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] && BooleanQ[index4]
RiemannTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_, index3_, index4_]["VanishingChernPontryaginScalarQ"] :=
Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, covariantRiemannTensor,
mixedRiemannTensor, leviCivitaTensor, contravariantLeviCivitaTensor}, If[Length[matrixRepresentation] == 4,
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
christoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]]; riemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> D[christoffelSymbols[[index[[1]],index[[2]],index[[4]]]],
newCoordinates[[index[[3]]]]] - D[christoffelSymbols[[index[[1]],index[[2]],index[[3]]]],
newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,index[[3]]]]*christoffelSymbols[[
#1,index[[2]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]] - Total[
(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ]; covariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,
index[[2]],index[[3]],index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]; mixedRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[2]],#1]]*
riemannTensor[[index[[1]],#1,index[[3]],index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]; leviCivitaTensor = Normal[LeviCivitaTensor[4]];
contravariantLeviCivitaTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*Inverse[matrixRepresentation][[#1[[3]],index[[3]]]]*
Inverse[matrixRepresentation][[#1[[4]],index[[4]]]]*leviCivitaTensor[[#1[[1]],#1[[2]],#1[[3]],
#1[[4]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]];
FullSimplify[Total[(covariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*contravariantLeviCivitaTensor[[
#1[[1]],#1[[2]],#1[[5]],#1[[6]]]]*mixedRiemannTensor[[#1[[3]],#1[[4]],#1[[5]],#1[[6]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 6]] == 0] === True, Indeterminate]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] && BooleanQ[index4]
RiemannTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_, index3_, index4_]["VanishingEulerScalarQ"] :=
Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, covariantRiemannTensor,
leviCivitaTensor, contravariantLeviCivitaTensor}, If[Length[matrixRepresentation] == 4,
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
christoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]]; riemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> D[christoffelSymbols[[index[[1]],index[[2]],index[[4]]]],
newCoordinates[[index[[3]]]]] - D[christoffelSymbols[[index[[1]],index[[2]],index[[3]]]],
newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,index[[3]]]]*christoffelSymbols[[
#1,index[[2]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]] - Total[
(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ]; covariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,
index[[2]],index[[3]],index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]; leviCivitaTensor = Normal[LeviCivitaTensor[4]];
contravariantLeviCivitaTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*Inverse[matrixRepresentation][[#1[[3]],index[[3]]]]*
Inverse[matrixRepresentation][[#1[[4]],index[[4]]]]*leviCivitaTensor[[#1[[1]],#1[[2]],#1[[3]],
#1[[4]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]];
FullSimplify[Total[(covariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*covariantRiemannTensor[[#1[[5]],
#1[[6]],#1[[7]],#1[[8]]]]*contravariantLeviCivitaTensor[[#1[[1]],#1[[2]],#1[[5]],#1[[6]]]]*
contravariantLeviCivitaTensor[[#1[[3]],#1[[4]],#1[[7]],#1[[8]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 8]] == 0] === True, Indeterminate]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] && BooleanQ[index4]
RiemannTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_, index3_, index4_]["RiemannFlatConditions"] :=
Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, fieldEquations},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
christoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]];
riemannTensor = Normal[SparseArray[(Module[{index = #1}, index -> D[christoffelSymbols[[index[[1]],index[[2]],
index[[4]]]], newCoordinates[[index[[3]]]]] - D[christoffelSymbols[[index[[1]],index[[2]],index[[3]]]],
newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,index[[3]]]]*christoffelSymbols[[
#1,index[[2]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]] -
Total[(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ];
fieldEquations = FullSimplify[Thread[Catenate[Catenate[Catenate[riemannTensor]]] ==
Catenate[Catenate[Catenate[ConstantArray[0, {Length[matrixRepresentation], Length[matrixRepresentation],
Length[matrixRepresentation], Length[matrixRepresentation]}]]]]]];
If[fieldEquations === True, {}, If[fieldEquations === False, Indeterminate,
If[Length[Select[fieldEquations, #1 === False & ]] > 0, Indeterminate,
DeleteDuplicates[Reverse /@ Sort /@ Select[fieldEquations, #1 =!= True & ]]]]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] && BooleanQ[index4]
RiemannTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_, index3_, index4_]["VanishingKretschmannScalarCondition"] :=
Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, covariantRiemannTensor,
contravariantRiemannTensor, fieldEquation},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
christoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]];
riemannTensor = Normal[SparseArray[(Module[{index = #1}, index -> D[christoffelSymbols[[index[[1]],index[[2]],
index[[4]]]], newCoordinates[[index[[3]]]]] - D[christoffelSymbols[[index[[1]],index[[2]],index[[3]]]],
newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,index[[3]]]]*christoffelSymbols[[
#1,index[[2]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]] -
Total[(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ]; covariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,
index[[2]],index[[3]],index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]; contravariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[2]],#1[[1]]]]*
Inverse[matrixRepresentation][[#1[[2]],index[[3]]]]*Inverse[matrixRepresentation][[#1[[3]],index[[4]]]]*
riemannTensor[[index[[1]],#1[[1]],#1[[2]],#1[[3]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
3]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]];
fieldEquation = FullSimplify[Total[(covariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*
contravariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]] == 0]; If[fieldEquation === False, Indeterminate,
fieldEquation]] /; SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] && BooleanQ[index4]
RiemannTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_, index3_, index4_]["VanishingChernPontryaginScalarCondition"] :=
Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, covariantRiemannTensor,
mixedRiemannTensor, leviCivitaTensor, contravariantLeviCivitaTensor, fieldEquation},
If[Length[matrixRepresentation] == 4, newMatrixRepresentation = matrixRepresentation /.
(#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
christoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]]; riemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> D[christoffelSymbols[[index[[1]],index[[2]],index[[4]]]],
newCoordinates[[index[[3]]]]] - D[christoffelSymbols[[index[[1]],index[[2]],index[[3]]]],
newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,index[[3]]]]*christoffelSymbols[[
#1,index[[2]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]] - Total[
(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ]; covariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,
index[[2]],index[[3]],index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]; mixedRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[2]],#1]]*
riemannTensor[[index[[1]],#1,index[[3]],index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]; leviCivitaTensor = Normal[LeviCivitaTensor[4]];
contravariantLeviCivitaTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*Inverse[matrixRepresentation][[#1[[3]],index[[3]]]]*
Inverse[matrixRepresentation][[#1[[4]],index[[4]]]]*leviCivitaTensor[[#1[[1]],#1[[2]],#1[[3]],
#1[[4]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]; fieldEquation =
FullSimplify[Total[(covariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*contravariantLeviCivitaTensor[[
#1[[1]],#1[[2]],#1[[5]],#1[[6]]]]*mixedRiemannTensor[[#1[[3]],#1[[4]],#1[[5]],#1[[6]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 6]] == 0]; If[fieldEquation === False, Indeterminate,
fieldEquation], Indeterminate]] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] &&
BooleanQ[index4]
RiemannTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_, index3_, index4_]["VanishingEulerScalarCondition"] :=
Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, covariantRiemannTensor,
leviCivitaTensor, contravariantLeviCivitaTensor, fieldEquation}, If[Length[matrixRepresentation] == 4,
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
christoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]]; riemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> D[christoffelSymbols[[index[[1]],index[[2]],index[[4]]]],
newCoordinates[[index[[3]]]]] - D[christoffelSymbols[[index[[1]],index[[2]],index[[3]]]],
newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,index[[3]]]]*christoffelSymbols[[
#1,index[[2]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]] - Total[
(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ]; covariantRiemannTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],#1]]*riemannTensor[[#1,
index[[2]],index[[3]],index[[4]]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]; leviCivitaTensor = Normal[LeviCivitaTensor[4]];
contravariantLeviCivitaTensor = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index[[1]],#1[[1]]]]*
Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*Inverse[matrixRepresentation][[#1[[3]],index[[3]]]]*
Inverse[matrixRepresentation][[#1[[4]],index[[4]]]]*leviCivitaTensor[[#1[[1]],#1[[2]],#1[[3]],
#1[[4]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]; fieldEquation =
FullSimplify[Total[(covariantRiemannTensor[[#1[[1]],#1[[2]],#1[[3]],#1[[4]]]]*covariantRiemannTensor[[#1[[5]],
#1[[6]],#1[[7]],#1[[8]]]]*contravariantLeviCivitaTensor[[#1[[1]],#1[[2]],#1[[5]],#1[[6]]]]*
contravariantLeviCivitaTensor[[#1[[3]],#1[[4]],#1[[7]],#1[[8]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 8]] == 0]; If[fieldEquation === False, Indeterminate,
fieldEquation], Indeterminate]] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && BooleanQ[index1] && BooleanQ[index2] && BooleanQ[index3] &&
BooleanQ[index4]
RiemannTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_], index1_,
index2_, index3_, index4_]["IndexContractions"] :=
Module[{newMatrixRepresentation, newCoordinates, christoffelSymbols, riemannTensor, newRiemannTensor},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
christoffelSymbols = Normal[SparseArray[
(Module[{index = #1}, index -> Total[((1/2)*Inverse[newMatrixRepresentation][[index[[1]],#1]]*
(D[newMatrixRepresentation[[#1,index[[3]]]], newCoordinates[[index[[2]]]]] + D[newMatrixRepresentation[[
index[[2]],#1]], newCoordinates[[index[[3]]]]] - D[newMatrixRepresentation[[index[[2]],index[[3]]]],
newCoordinates[[#1]]]) & ) /@ Range[Length[newMatrixRepresentation]]]] & ) /@
Tuples[Range[Length[newMatrixRepresentation]], 3]]];
riemannTensor = Normal[SparseArray[(Module[{index = #1}, index -> D[christoffelSymbols[[index[[1]],index[[2]],
index[[4]]]], newCoordinates[[index[[3]]]]] - D[christoffelSymbols[[index[[1]],index[[2]],index[[3]]]],
newCoordinates[[index[[4]]]]] + Total[(christoffelSymbols[[index[[1]],#1,index[[3]]]]*christoffelSymbols[[
#1,index[[2]],index[[4]]]] & ) /@ Range[Length[newMatrixRepresentation]]] -
Total[(christoffelSymbols[[index[[1]],#1,index[[4]]]]*christoffelSymbols[[#1,index[[2]],index[[3]]]] & ) /@
Range[Length[newMatrixRepresentation]]]] & ) /@ Tuples[Range[Length[newMatrixRepresentation]], 4]]] /.
(ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ];
If[(index1 === True && index2 === True && index3 === True && index4 === True) ||
(index1 === False && index2 === False && index3 === False && index4 === False), Association[],
If[index1 === True && index2 === False && index3 === False && index4 === False,
newRiemannTensor = Normal[SparseArray[(Module[{index = #1}, index -> Total[(matrixRepresentation[[index[[1]],
#1[[1]]]]*Inverse[matrixRepresentation][[index[[2]],#1[[2]]]]*Inverse[matrixRepresentation][[#1[[3]],
index[[3]]]]*Inverse[matrixRepresentation][[#1[[4]],index[[4]]]]*riemannTensor[[#1[[1]],#1[[2]],
#1[[3]],#1[[4]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 4]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]; Association[Subsuperscript["\[FormalCapitalR]", "\[FormalLambda]", "\[FormalLambda]\[FormalMu]\[FormalNu]"] ->
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newRiemannTensor[[#1,#1,First[index],
Last[index]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], Subsuperscript["\[FormalCapitalR]", "\[FormalLambda]", "\[FormalSigma]\[FormalLambda]\[FormalNu]"] ->
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newRiemannTensor[[#1,First[index],#1,
Last[index]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]], Subsuperscript["\[FormalCapitalR]", "\[FormalLambda]", "\[FormalSigma]\[FormalMu]\[FormalLambda]"] ->
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newRiemannTensor[[#1,First[index],Last[index],
#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
2]]]], If[index1 === False && index2 === True && index3 === False && index4 === False,
newRiemannTensor = Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[
#1[[1]],index[[3]]]]*Inverse[matrixRepresentation][[#1[[2]],index[[4]]]]*riemannTensor[[index[[1]],
index[[2]],#1[[1]],#1[[2]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]; Association[Subsuperscript["\[FormalCapitalR]", "\[FormalLambda]", "\[FormalLambda]\[FormalMu]\[FormalNu]"] ->
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newRiemannTensor[[#1,#1,First[index],
Last[index]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@ Tuples[Range[
Length[matrixRepresentation]], 2]]], Subsuperscript["\[FormalCapitalR]", "\[FormalLambda]", "\[FormalRho]\[FormalLambda]\[FormalNu]"] ->
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newRiemannTensor[[First[index],#1,#1,
Last[index]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@ Tuples[Range[
Length[matrixRepresentation]], 2]]], Subsuperscript["\[FormalCapitalR]", "\[FormalLambda]", "\[FormalRho]\[FormalMu]\[FormalLambda]"] ->
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newRiemannTensor[[First[index],#1,Last[index],
#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
2]]]], If[index1 === False && index2 === False && index3 === True && index4 === False,
newRiemannTensor = Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[
index[[2]],#1[[1]]]]*Inverse[matrixRepresentation][[#1[[2]],index[[4]]]]*riemannTensor[[index[[1]],
#1[[1]],index[[3]],#1[[2]]]] & ) /@ Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 4]]]; Association[Subsuperscript["\[FormalCapitalR]", "\[FormalLambda]", "\[FormalLambda]\[FormalSigma]\[FormalNu]"] ->
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newRiemannTensor[[#1,First[index],#1,
Last[index]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@ Tuples[
Range[Length[matrixRepresentation]], 2]]], Subsuperscript["\[FormalCapitalR]", "\[FormalLambda]", "\[FormalRho]\[FormalLambda]\[FormalNu]"] ->
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newRiemannTensor[[First[index],#1,#1,
Last[index]]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@ Tuples[
Range[Length[matrixRepresentation]], 2]]], Subsuperscript["\[FormalCapitalR]", "\[FormalLambda]", "\[FormalRho]\[FormalSigma]\[FormalLambda]"] ->
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newRiemannTensor[[First[index],Last[index],#1,
#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@ Tuples[
Range[Length[matrixRepresentation]], 2]]]], If[index1 === False && index2 === False &&
index3 === False && index4 === True, newRiemannTensor = Normal[SparseArray[(Module[{index = #1},
index -> Total[(Inverse[matrixRepresentation][[index[[2]],#1[[1]]]]*Inverse[matrixRepresentation][[
#1[[2]],index[[3]]]]*riemannTensor[[index[[1]],#1[[1]],#1[[2]],index[[4]]]] & ) /@
Tuples[Range[Length[matrixRepresentation]], 2]]] & ) /@ Tuples[Range[Length[matrixRepresentation]],
4]]]; Association[Subsuperscript["\[FormalCapitalR]", "\[FormalLambda]", "\[FormalLambda]\[FormalSigma]\[FormalMu]"] ->
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newRiemannTensor[[#1,First[index],Last[index],
#1]] & ) /@ Range[Length[matrixRepresentation]]]] & ) /@ Tuples[Range[Length[
matrixRepresentation]], 2]]], Subsuperscript["\[FormalCapitalR]", "\[FormalLambda]", "\[FormalRho]\[FormalLambda]\[FormalMu]"] ->