calculate the Killing tensors of a specified rank for a given metric or connection - Maple Programming Help

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Tensor[KillingTensors] - calculate the Killing tensors of a specified rank for a given metric or connection

Calling Sequences

     KillingTensors(g, p options)

     KillingTensors(C, p options)

Parameters

     g         - a covariant metric tensor on a manifold M

     p         - a positive integer

     C         - an affine connection on a manifold M

     options   - any of the following keywords arguments: ansatz, unknowns, auxiliaryequations, coefficientvariables, parameters, output 

 

Description

Examples

Description

• 

 Let denote covariant differentiation with respect to the given metric g or connection C.A covariant symmetric tensor field T of rank p is called a Killing tensor if (iTjk... l) = 0.

• 

The program KillingTensors generates the defining 1st order partial differential equations for a Killing tensor of rank p and uses pdsolve to find the solution to these equations. An empty list is returned if there are no Killing tensors. If pdsolve is unable to solve these equations, NULL is returned.

• 

The keyword argument coefficientvariables = x1, x2, ... , xk allows the user to specify the coefficient functions in the Killing tensor T as functions of the variables x1, x2, ... , xk .

• 

The exact form of the Killing tensor T can be specified with the keyword argument ansatz = T. For example, if the coordinates on the underlying manifold are x, y, z and T1, T2 are defined symmetric tensors, then one may solve for Killing tensors of the form T = fy, zT1 + gy,zT2 . In this situation the unknown functions must be explicitly specified with the keyword argument unknowns, for example, unknowns = fy,z, gy,z.

• 

When using the keyword argument ansatz, additional algebraic or differential conditions may be imposed upon the unknowns using the keyword argument auxiliaryequations = EqList. Here EqList  is a list of the auxiliary equations to be added to the Killing tensor equations.

• 

If the metric g or connection C depends upon a number of unspecified parameters (either constants or functions), then the keyword argument parameters= ParList,where ParList is the list of parameters, will invoke case-splitting with respect to these parameters. Special values of the parameters, where either the number or the explicit form of the Killing tensors changes, are calculated.

• 

With keyword argument output = pde,the defining partial differential equations for the Killing tensors are returned. The option output = general returns the general Killing tensor in terms of a number of arbitrary constants _C1, _C2 ... . The option output = list returns a list of tensors which form a basis for the solution space. The default value of this keyword argument is output = list.

• 

This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form KillingTensors(...) only after executing the commands with(DifferentialGeometry), with(Tensor) in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-KillingTensors(...).

Examples

withDifferentialGeometry:withTensor:

 

Example 1.

Find the Killing tensors of the metric g to order 3. This metric appears in Darboux, Theorie Generale des Surfaces III, page 81.

DGsetupx,y,M

frame name: M

(2.1)
M > 

g1evalDG1ydx&tdx+1xdy&tdy

_DGtensor,M,cov_bas,cov_bas,,1,1,1y,2,2,1x

(2.2)
M > 

K1KillingTensorsg1,1

(2.3)
M > 

K2KillingTensorsg1,2

_DGtensor,M,cov_bas,cov_bas,,1,1,1y,2,2,1x

(2.4)
M > 

K3KillingTensorsg1,3

_DGtensor,M,cov_bas,cov_bas,cov_bas,,1,1,1,1y3,2,2,2,1x3

(2.5)

 

Example 2.

We use the keyword argument coefficientvariables to find the rank 3 Killing tensors for the metric g2 which are functions of y alone.

M > 

DGsetupx,y,z,M

frame name: M

(2.6)
M > 

g2evalDGdx&tdx+dy&tdy+xdz&tdz

_DGtensor,M,cov_bas,cov_bas,,1,1,1,2,2,1,3,3,x

(2.7)
M > 

KillingTensorsg2,3,coefficientvariables=y

_DGtensor,M,cov_bas,cov_bas,cov_bas,,2,2,2,1

(2.8)

 

Example 3.

We use the keyword arguments ansatz and unknowns to find the rank 2 Killing tensors for the metric g3 which are independent of dy .

M > 

DGsetupx,y,z,M

frame name: M

(2.9)
M > 

g3evalDGdx&tdx+dy&tdy+xdz&tdz

g3:=_DGtensor,M,cov_bas,cov_bas,,1,1,1,2,2,1,3,3,x

(2.10)
M > 

TevalDGAx,y,zdx&tdx+Bx,y,zdx&sdz+Cx,y,zdz&tdz

T:=_DGtensor,M,cov_bas,cov_bas,,1,1,Ax,y,z,1,3,Bx,y,z2,3,1,Bx,y,z2,3,3,Cx,y,z

(2.11)
M > 

KillingTensorsg3,2,ansatz=T,unknowns=Ax,y,z,Bx,y,z,Cx,y,z

_DGtensor,M,cov_bas,cov_bas,,1,1,1,3,3,x,_DGtensor,M,cov_bas,cov_bas,,3,3,x2

(2.12)

 

Example 4.

We use the keyword arguments ansatz, unknowns and auxiliaryequations find the rank 2 Killing tensors for the metric g4 which are invariant under rotations in the x y plane.

M > 

DGsetupx,y,z,M

frame name: M

(2.13)
M > 

g4evalDGdx&tdx+dy&tdy+x2+y2dz&tdz

g4:=_DGtensor,M,cov_bas,cov_bas,,1,1,1,2,2,1,3,3,x2+y2

(2.14)

 

We use the commands GenerateSymmmetricTensors and DGzip to construct the general rank 2 symmetric tensor T on M.

M > 

T0GenerateSymmetricTensorsdx,dy,dz,2

T0:=_DGtensor,M,cov_bas,cov_bas,,1,1,1,_DGtensor,M,cov_bas,cov_bas,,1,2,12,2,1,12,_DGtensor,M,cov_bas,cov_bas,,1,3,12,3,1,12,_DGtensor,M,cov_bas,cov_bas,,2,2,1,_DGtensor,M,cov_bas,cov_bas,,2,3,12,3,2,12,_DGtensor,M,cov_bas,cov_bas,,3,3,1

(2.15)
M > 

varsA1,A2,A3,A4,A5,A6x,y,z

vars:=A1x,y,z,A2x,y,z,A3x,y,z,A4x,y,z,A5x,y,z,A6x,y,z

(2.16)
M > 

TDGzipvars,T0,plus

T:=_DGtensor,M,cov_bas,cov_bas,,1,1,A1x,y,z,1,2,A2x,y,z2,1,3,A3x,y,z2,2,1,A2x,y,z2,2,2,A4x,y,z,2,3,A5x,y,z2,3,1,A3x,y,z2,3,2,A5x,y,z2,3,3,A6x,y,z

(2.17)

 

The vector field X is the infinitesimal generator for rotations in the x y plane.

M > 

XevalDGyD_xxD_y

X:=_DGvector,M,,1,y,2,x

(2.18)

 

We use the commands LieDerivative and DGinfo to find the conditions SymmetryEq under which T is rotationally invariant.

M > 

LDLieDerivativeX,T:

M > 

SymmetryEqTools:-DGinfoLD,CoefficientSet

SymmetryEq:=yxA6x,y,zxyA6x,y,z,A2x,y,z+yxA1x,y,zxyA1x,y,z,A2x,y,z+yxA4x,y,zxyA4x,y,z,12A3x,y,z+12yxA5x,y,z12xyA5x,y,z,12A5x,y,z+12yxA3x,y,z12xyA3x,y,z,A4x,y,z+A1x,y,z+12yxA2x,y,z12xyA2x,y,z

(2.19)

 

We find that that there are 4 rotationally invariant, rank 2 Killing tensors for the metric g4.

M > 

InvKTKillingTensorsg4,ansatz=T,unknowns=vars,auxiliaryequations=SymmetryEq

InvKT:=_DGtensor,M,cov_bas,cov_bas,,1,1,12,2,2,12,3,3,x22y22,_DGtensor,M,cov_bas,cov_bas,,3,3,x2+y22,_DGtensor,M,cov_bas,cov_bas,,1,3,yx2+y22,2,3,xx2+y22,3,1,yx2+y22,3,2,xx2+y22,_DGtensor,M,cov_bas,cov_bas,,1,1,y2,1,2,yx,2,1,yx,2,2,x2

(2.20)
M > 

nopsInvKT

4

(2.21)

 

 

Example 5.

We wish to determine the rank 2 Killing tensors for the metric g5 =zm +1 dx dx +1zm  +1 dy dy + dz dz for varying values of m,excluding the case m=0. Because the parameter m does not appear as a rational function in g5, it is helpful to re-write the metric as g5=az +1 dx dx +1az+1 dy dy + dz dz , where azsatisfies the differential equation z a'z = m az.  

M > 

DGsetupx,y,z,M

frame name: M

(2.22)
M > 

g5evalDGaz+1dx&tdx+1az+1dy&tdy+dz&tdz

_DGtensor,M,cov_bas,cov_bas,,1,1,az+1,2,2,1az+1,3,3,1

(2.23)
M > 

KTKillingTensorsg5,2,parameters=m,az,auxiliaryequations=m0,az0,zdiffaz,z=maz:

 

With the keyword argument parameters, the command KillingTensors returns a sequence of lists of Killing tensors and, as the last element of the sequence, the possible exceptional parameter values. For this example, the exceptional values of m are:

M > 

CasesKT1

m=1,az=_C1z,m=_C1,az=z_C1_C2

(2.24)

 

We see there are 2 cases.

Case 1. m = 1

M > 

KT1

_DGtensor,M,cov_bas,cov_bas,,1,1,_C1z+1,2,2,_C1z_C1z+12,3,3,1,_DGtensor,M,cov_bas,cov_bas,,2,2,1_C1z+12,_DGtensor,M,cov_bas,cov_bas,,1,2,12,2,1,12,_DGtensor,M,cov_bas,cov_bas,,1,1,_C1z+12

(2.25)

 

Case 2. m =  C1  (the generic case)

M > 

KT3

m=1,az=_C1z,m=_C1,az=z_C1_C2

(2.26)

 

Example 6.

With the keyword argument output = "pde", the defining partial differential equations for the Killing tensor are returned.

M > 

DGsetupx,y,z,M

frame name: M

(2.27)
M > 

g6evalDGdx&tdx+dy&tdy+xdz&tdz

_DGtensor,M,cov_bas,cov_bas,,1,1,1,2,2,1,3,3,x

(2.28)
M > 

TevalDGAx,ydx&tdx+Bx,ydy&tdy

_DGtensor,M,cov_bas,cov_bas,,1,1,Ax,y,2,2,Bx,y

(2.29)
M > 

KillingTensorsg6,ansatz=T,unknowns=Ax,y,Bx,y

_DGtensor,M,cov_bas,cov_bas,,2,2,1

(2.30)
M > 

KillingTensorsg6,ansatz=T,unknowns=Ax,y,Bx,y,output=pde

0,13Ax,y,13yAx,y,13xBx,y,xAx,y,yBx,y

(2.31)

 

Example 7.

We compute the Killing tensors for a connection. We use the keyword argument output = "general" to obtain the result as a single tensor depending on constants _C1, _C2, ...  .

M > 

DGsetupx,y,M

frame name: M

(2.32)
M > 

CConnectionyD_y&tdx&tdy+D_y&tdy&tdx

_DGconnection,M,con_bas,cov_bas,cov_bas,,2,1,2,y,2,2,1,y

(2.33)
M > 

KillingTensorsC,2,output=general

_DGtensor,M,cov_bas,cov_bas,,1,1,_C1y4+_C2y2+_C3,1,2,_C1y2+12_C2,2,1,_C1y2+12_C2,2,2,_C1

(2.34)

 

Example 8.

The following metric g8 appears in the PhD thesis of R. P. Delong. We are able to explicitly compute all Killing tensors to order 4. The explicit lists are very long and so we simply display the number of Killing tensors at each order.

M > 

DGsetupx,y,z,M

frame name: M

(2.35)
M > 

g8evalDGxdx&sdy+xdz&tdz

_DGtensor,M,cov_bas,cov_bas,,1,2,12x,2,1,12x,3,3,x

(2.36)
M > 

K1KillingTensorsg8,1:

M > 

K2KillingTensorsg8,2:

M > 

K3KillingTensorsg8,3:

M > 

K4KillingTensorsg8,4:

M > 

nopsK1,nopsK2,nopsK3,nopsK4

4,14,32,69

(2.37)

See Also

DifferentialGeometry

Tensor

KillingVectors

KillingYanoTensors

KillingSpinors