I want to define a symplectic manifold with symplectic form $\Omega_{[ab]}$ with $\Omega_{ab}\Omega^{ac} = \delta^c_b$, but without having a metric.
I define a manifold and an antisymmetric (2,0) tensor.
<< xAct`xTensor`;
DefManifold[sympl, 8, {a, b, c, d, e, f, g, h, i, j, k, l}];
DefTensor[Omega[a, b], sympl, Antisymmetric[{a, b}]];
To implement the relation to the inverse I define a rule
omegarule =
MakeRule[{ Omega[-a, -b] Omega[a, c], delta[c, -b]},
PatternIndices -> All];
But to define the rule I need to make use of the delta[]
function and this is somehow associated to the metric. Thus sometimes Mathematica will complain and tell me:
E.g. I define
DefTensor[X[a, b], sympl, Symmetric[{a, b}]];
exp1 = X[a, e] Omega[b, f] Omega[c, g] Omega[d, h] -
X[a, e] Omega[b, c] Omega[f, g] Omega[d, h]
exp1Asym =
24 Antisymmetrize[
24 Antisymmetrize[exp1, {a, b, c, d}], {e, f, g, h}]
But
exp1Asym Omega[-a, -b] // ToCanonical
% /. omegarule
Will produce the error.
But
Antisymmetrize[(X[a, e] Omega[b, f] Omega[c, g]), {b,
c}] Omega[-f, -h] // ToCanonical
% /. omegarule
Will not produce the error, althought it is the same situation.
Plus, if I do not use the %
the rule is not being applied in these cases at all. But in smaller cases it is.
How do I treat symplectic geometry properly? If I do define a metric it will sometimes lower or raise indices of the symplectic form, which does not make sense.