CheckKillingTensor - Maple Help
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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

Description

• 

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.

Examples

with(DifferentialGeometry): with(Tensor):

 

Example 1.

Check that  is a Killing tensor for the metric .

DGsetup([x, y], M):

M > 

g := evalDG((1/y)*dx &t dx + 1/x*dy &t dy);

(2.1)
M > 

K1 := evalDG((1/y^3)*dx &t dx &t dx - (1/x^3)*dy &t dy &t dy);

(2.2)
M > 

CheckKillingTensor(g, K1);

(2.3)

 

Example 2.

Determine the equations for  and that must be satisfied for  to be a Killing tensor for the metric  from Example 1.

M > 

K2 := evalDG(A(y)*dx &t dx &t dx + B(x)*dy &t dy &t dy);

(2.4)
M > 

P := CheckKillingTensor(g, K2);

(2.5)
M > 

Tools:-DGinfo(P, "CoefficientSet");

(2.6)

See Also

DifferentialGeometry

Tensor

CovariantDerivative

SymmetrizeIndices

 


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