check that a tensor is the Killing tensor for a given metric or connection - 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

• 

This program computes the symmetrized covariant derivative of the symmetric covariant tensor T with respect to the Christoffel connection of the metric g or the given connection C, that is, it computes the Killing tensor equation (aTbcd ...).

• 

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 K1 is a Killing tensor for the metric g.

DGsetup([x, y], M):

M > 

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

g:=1ydxdx+1xdydy

(2.1)
M > 

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

K1:=1y3dxdxdx1x3dydydy

(2.2)
M > 

CheckKillingTensor(g, K1);

0dxdxdxdx

(2.3)

 

Example 2.

Determine the equations for Ay and Bx that must be satisfied for K2 to be a Killing tensor for the metric g from Example 1.

M > 

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

K2:=Aydxdxdx+Bxdydydy

(2.4)
M > 

P := CheckKillingTensor(g, K2);

P:=ⅆⅆyAyy+3Ay4ydxdxdxdy+ⅆⅆyAyy+3Ay4ydxdxdydxx3Bx+y3Ay4x2y2dxdxdydy+ⅆⅆyAyy+3Ay4ydxdydxdxx3Bx+y3Ay4x2y2dxdydxdyx3Bx+y3Ay4x2y2dxdydydx+Bxx+3Bx4xdxdydydy+ⅆⅆyAyy+3Ay4ydydxdxdxx3Bx+y3Ay4x2y2dydxdxdyx3Bx+y3Ay4x2y2dydxdydx+Bxx+3Bx4xdydxdydyx3Bx+y3Ay4x2y2dydydxdx+Bxx+3Bx4xdydydxdy+Bxx+3Bx4xdydydydx

(2.5)
M > 

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

14ⅆⅆyAyy+3Ayy,14ⅆⅆxBxx+3Bxx,14x3Bx+y3Ayx2y2

(2.6)

See Also

DifferentialGeometry, Tensor, CovariantDerivative, SymmetrizeIndices


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