Tensor[CheckKillingTensor] - check that a tensor is the Killing tensor for a given metric or connection
Calling Sequences
CheckKillingTensor(g, T)
CheckKillingTensor(C, T)
Parameters
g - a covariant metric tensor on a manifold M
T - a symmetric covariant tensor on M, or a list of such
C - an affine connection on a manifold M
Description
Examples
This program computes the symmetrized covariant derivative of the symmetric covariant tensor with respect to the Christoffel connection of the metric or the given connection , that is, it computes the Killing tensor equation .
This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form CheckKillingTensor(...) only after executing the commands with(DifferentialGeometry), with(Tensor) in that order. It can always be used in the long form DifferentialGeometry:-Tensor:CheckKillingTensor.
with(DifferentialGeometry): with(Tensor):
Example 1.
Check that is a Killing tensor for the metric .
DGsetup([x, y], M):
g := evalDG((1/y)*dx &t dx + 1/x*dy &t dy);
K1 := evalDG((1/y^3)*dx &t dx &t dx - (1/x^3)*dy &t dy &t dy);
CheckKillingTensor(g, K1);
Example 2.
Determine the equations for and that must be satisfied for to be a Killing tensor for the metric from Example 1.
K2 := evalDG(A(y)*dx &t dx &t dx + B(x)*dy &t dy &t dy);
P := CheckKillingTensor(g, K2);
Tools:-DGinfo(P, "CoefficientSet");
See Also
DifferentialGeometry
Tensor
CovariantDerivative
SymmetrizeIndices
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