Problem displaying Killing equations

What I'm trying to do is to compute the Killing equations and then solve them using DSolve to determine their solutions. I have a problem understanding where my Mathematica program doesn't work properly as it seems like it doesn't substitute \ [Micro], \ [Nu] and \ [Delta] properly... For simplicity I adapted my program to compute the Killing equations for a 2-sphere where the Killing equations are well known.

The general expression for the Killing equations I decided to work with is of the form:

The equations that I should obtain for the 2-sphere should be of the form:

    (*coordinates initialisation*)
Clear[coord, metric, inversemetric, affine, killing, t, r, \[Theta], \
\[Phi]]

n = 2

(*The metric and inverse metric*)
coord = {\[Theta], \[Phi]}
metric = {{1, 0}, {0, sin[\[Theta]]*sin[\[Theta]]}}

inversemetric = Simplify[Inverse[metric]]

(*Calculating the Christoffel symbols*)
affine :=
affine = Simplify[
Table[(1/2)*
Sum[(inversemetric[[i, s]])*(D[metric[[s, j]], coord[[k]]] +
D[metric[[s, k]], coord[[j]]] -
D[metric[[j, k]], coord[[s]]]), {s, 1, n}], {i, 1, n}, {j, 1,
n}, {k, 1, n}]]

(*Dysplaying non-null Christoffel symbols*)
listaffine :=
Table[If[UnsameQ[affine[[i, j, k]],
0], {ToString[\[CapitalGamma][i, j, k]], affine[[i, j, k]]}], {i,
1, n}, {j, 1, n}, {k, 1, n}]
TableForm[Partition[DeleteCases[Flatten[listaffine], Null], 2],
TableSpacing -> {2, 2}]

(*Killing equations*)
killing :=
killing =
Simplify[Table[
D[Subscript[\[Xi], coord[[\[Micro]]]][\[Theta], \[Phi]],
coord[[\[Nu]]]] +
D[Subscript[\[Xi], coord[[\[Nu]]]][\[Theta], \[Phi]],
coord[[\[Micro]]]] -
2*affine[[\[Delta], \[Micro], \[Nu]]]*
Subscript[\[Xi],
coord[[\[Delta]]]][\[Theta], \[Phi]], {\[Micro], 1, n}, {\[Nu],
1, n}, {\[Delta], 1, n}]]

(*Dysplaying Killing equations*)
listkilling :=
Table[If[UnsameQ[killing[[\[Micro], \[Nu], \[Delta]]],
0], {killing[[\[Micro], \[Nu], \[Delta]]], "=  0"}], {\[Micro], 1,
n}, {\[Nu], 1, n}, {\[Delta], 1, n}]
TableForm[Partition[DeleteCases[Flatten[listkilling], Null], 2],
TableSpacing -> {2, 2}]


The results my code gives are of the form:

where I believe there are some obvious differences. First of all there is a suplementary equation (the third one displayer above) and in the first two the derivative si carried for both coordinates instead of only \ [Theta]. Any opinions? Thank you in advance!

PS: How can I get rid of that annoying 'Subscript' and display it properly?

• I find it somewhat odd that you use \[Micro] instead of \[Mu]... Moreover affine := affine = ... is odd, to. Why don't you just use affine = ...? – Henrik Schumacher Dec 27 '18 at 9:39
• @HenrikSchumacher \ [Micro] is equivalent to \ [Mu] it seems so it doesn't bother me. I used := as I previously got some errors. In this way the defined function/equation remains unevaluated until I substitute the subsripts/ other parameters. – Vlad G Dec 27 '18 at 14:50

(*coordinates initialisation*)Clear[coord, metric, inversemetric, \
affine, killing, t, r, \[Theta], \[Phi]]
n = 2
(*The metric and inverse metric*)
coord = {\[Theta], \[Phi]}
metric = {{1, 0}, {0, Sin[\[Theta]]*Sin[\[Theta]]}}

inversemetric = Simplify[Inverse[metric]]
(*Calculating the Christoffel symbols*)
affine :=
affine = Simplify[
Table[(1/2)*
Sum[(inversemetric[[i, s]])*(D[metric[[s, j]], coord[[k]]] +
D[metric[[s, k]], coord[[j]]] -
D[metric[[j, k]], coord[[s]]]), {s, 1, n}], {i, 1, n}, {j, 1,
n}, {k, 1, n}]]
(*Dysplaying non-null Christoffel symbols*)listaffine :=
Table[If[UnsameQ[affine[[i, j, k]],
0], {ToString[\[CapitalGamma][i, j, k]], affine[[i, j, k]]}], {i,
1, n}, {j, 1, n}, {k, 1, n}]
TableForm[Partition[DeleteCases[Flatten[listaffine], Null], 2],
TableSpacing -> {2, 2}]
(*Killing equations*)
killing :=
killing =
Simplify[Table[
D[Subscript[\[Xi], \[Micro]][\[Theta], \[Phi]], coord[[\[Nu]]]] +
D[Subscript[\[Xi], \[Nu]][\[Theta], \[Phi]],
coord[[\[Micro]]]] -
2*affine[[\[Delta], \[Micro], \[Nu]]]*
Subscript[\[Xi], \[Delta]][\[Theta], \[Phi]], {\[Micro], 1,
n}, {\[Nu], 1, n}, {\[Delta], 1, n}]]

(*Dysplaying Killing equations*)listkilling :=
Table[If[UnsameQ[killing[[\[Micro], \[Nu], \[Delta]]],
0], {killing[[\[Micro], \[Nu], \[Delta]]], "=  0"}], {\[Micro], 1,
n}, {\[Nu], 1, n}, {\[Delta], 1, n}]
TableForm[Partition[DeleteCases[Flatten[listkilling], Null], 2],
TableSpacing -> {2, 2}]


• Yes! Thank you sir, I must recognize that I wanted to put that coord[[ ]] in my subscripts but it messed up the things. I am still puzzled by the 3rd and 5th equation. I did the calculations by myself/by hand and substituting the subscripts indeed gives this equation too though non of the referrences I worked with mentioned a 4th equation. Thank you again!!! – Vlad G Dec 27 '18 at 14:53
• You're welcome! – Alex Trounev Dec 27 '18 at 14:55