calculate the recurrent tensors with respect to a given metric or connection - Maple Programming Help

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Tensor[RecurrentTensors] - calculate the recurrent tensors with respect to a given metric or connection

Calling Sequences

    RecurrentTensors(g, T, options)

    RecurrentTensors( Γ, T, options)

Parameters

  g        - a metric tensor on a manifold M

  Γ        - a connection, either an affine connection on M or a connection on a vector bundle E over M 

  T        - a list of vector fields, differential forms or tensors (all of the same type)

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

 

 

Description

Examples

Description

• 

Let  denote covariant differentiation with respect to the given connection Γ, or with respect to the Christoffel connection defined by the metric g. A tensor field T is called recurrent with respect to  if there exists a 1-form α such thatT = T α. The form α is called the eigenform for T. 

• 

 Let T=T1, T2, .... , Tp be a list of p tensor fields, all of the same covariant-contravariant type and let T = f1T1 + f2 T2 +  + fpTp, where the coefficients fi are arbitrary functions on the underlying manifold. The command RecurrentTensor generates the system of first order PDE in the unknowns fi and the components of α from the tensor equation T = T α and uses pdsolve to find the solutions to these PDE. Note that this a non-linear system of PDE.

• 

 If T  is a recurrent tensor and f  is a smooth function on M, then f T  is also a recurrent tensor. The algorithm used by the RecurrentTensors program is to first look for recurrent tensors of the form  T = T1 + f2 T2 +  + fpTp f1 =1, then recurrent tensors of the form  T = T2 +  + fpTp f1 =0, f2  = 1 and so on. Thus the leading coefficients (with respect to the basis T) in the output are always 1.

• 

If T = T α and the 1-form α is exact, that is, α= dg, then the tensor S= expgT  is covariantly constant: S= 0. Covariantly constant tensors can be computed with the command CovariantlyConstantTensors.  By convention, recurrent tensors whose eigenforms are closed (ⅆα =0) are not included in the default output to the command RecurrentTensor.

• 

The output from the RecurrentTensor command is a sequence of 2 lists. The first is a list of recurrent tensors and the second is the list of associated eigenforms.

• 

 The coefficient functions fi are taken to be functions of all the coordinate variables. The keyword argument coefficientvariables = x1 , x2, ... , xk allows the user to specify the coefficients functions fi  as functions of the variables x1 , x2, ... , xk .

• 

If the metric g or the connection Γ 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 recurrent tensors changes, are calculated.

• 

With keyword argument output = pde, the defining partial differential equations for the recurrent tensors are returned. With output = all ,all recurrent tensors (including those with closed eigenforms) are returned.

• 

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

Examples

withDifferentialGeometry:withTensor:

 

Example 1.

We find the recurrent 2 forms for a metric g, defined on a 3-dimensional manifold.

DGsetupx,y,z,M

frame name: M

(2.1)

gevalDGz2dx&tdx+dy&tdyxdz&tdz

_DGtensor,M,cov_bas,cov_bas,,1,1,z2,2,2,1,3,3,x

(2.2)

 

We use the command GenerateForms to generate a basis Ω for the space of degree 2 forms.

M > 

ΩTools:-GenerateFormsdx,dy,dz,2

_DGform,M,2,1,2,1,_DGform,M,2,1,3,1,_DGform,M,2,2,3,1

(2.3)

 

There are 2 recurrent 2-forms.

M > 

R,αRecurrentTensorsg,Ω

_DGform,M,2,1,2,1,2,3,xz,_DGform,M,2,1,2,1,2,3,xz,_DGform,M,1,1,1x,3,122x+1xz,_DGform,M,1,1,1x,3,122x1xz

(2.4)

 

We can check these answers by back-substituting into the recurrent tensor equation. To this end, we need the Christoffel connection for the metric g.

M > 

CChristoffelg

_DGconnection,M,con_bas,cov_bas,cov_bas,,1,1,3,1z,1,3,1,1z,1,3,3,12z2,3,1,1,zx,3,1,3,12x,3,3,1,12x

(2.5)

 

The first 2-form in the list R is recurrent.

M > 

CovariantDerivativeR1,C&minusR1&tensorα1

_DGtensor,M,cov_bas,cov_bas,cov_bas,,1,1,1,0

(2.6)

 

The second 2-form in the list R is recurrent.

M > 

CovariantDerivativeR2,C&minusR2&tensorα2

_DGtensor,M,cov_bas,cov_bas,cov_bas,,1,1,1,0

(2.7)

 

Example 2.

We find the recurrent rank 2 symmetric tensors for the metric g from Example 1.

First we use the command GenerateSymmetricTensors to generate a basis S for the space of rank 2 symmetric tensors.

M > 

SGenerateSymmetricTensorsdx,dy,dz,2

_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

(2.8)

 

There are 4 recurrent tensors.

M > 

R2,α2RecurrentTensorsg,S

_DGtensor,M,cov_bas,cov_bas,,1,1,1,1,3,xz,3,1,xz,3,3,xz2,_DGtensor,M,cov_bas,cov_bas,,1,1,1,1,3,xz,3,1,xz,3,3,xz2,_DGtensor,M,cov_bas,cov_bas,,1,2,12,2,1,12,2,3,12xz,3,2,12xz,_DGtensor,M,cov_bas,cov_bas,,1,2,12,2,1,12,2,3,12xz,3,2,12xz,_DGform,M,1,1,2x,3,2x+1xz,_DGform,M,1,1,2x,3,2x1xz,_DGform,M,1,1,1x,3,122x+1xz,_DGform,M,1,1,1x,3,122x1xz

(2.9)
M > 

nopsR2

4

(2.10)

 

There are two additional recurrent tensors which correspond to covariantly constant tensors and hence have a closed eigenform. We can see this with the option output = "all".

M > 

R2a,alpha2aRecurrentTensorsg,S,output=all

_DGtensor,M,cov_bas,cov_bas,,1,1,1,2,2,_C1z2,3,3,xz2,_DGtensor,M,cov_bas,cov_bas,,1,1,1,1,3,xz,3,1,xz,3,3,xz2,_DGtensor,M,cov_bas,cov_bas,,1,1,1,1,3,xz,3,1,xz,3,3,xz2,_DGtensor,M,cov_bas,cov_bas,,1,2,12,2,1,12,2,3,12xz,3,2,12xz,_DGtensor,M,cov_bas,cov_bas,,1,2,12,2,1,12,2,3,12xz,3,2,12xz,_DGtensor,M,cov_bas,cov_bas,,2,2,1,_DGform,M,1,3,2z,_DGform,M,1,1,2x,3,2x+1xz,_DGform,M,1,1,2x,3,2x1xz,_DGform,M,1,1,1x,3,122x+1xz,_DGform,M,1,1,1x,3,122x1xz,_DGform,M,1,1,0

(2.11)
M > 

nopsR2a

6

(2.12)

 

Note that the 1st and last entries in α2a are closed 1-forms. This implies that there are 2 covariantly constant tensors. We can check this directly using the CovariantlyConstantTensors command.

M > 

CovariantlyConstantTensorsg,S

_DGtensor,M,cov_bas,cov_bas,,1,1,z2,3,3,x,_DGtensor,M,cov_bas,cov_bas,,2,2,1

(2.13)

 

Example 3.

In this example we consider a metric g3 which depends upon arbitrary parameters a, b. We find that there are additional recurrent vectors when a=0 or b=0. 

M > 

g3evalevalDGdx&tdx+dy&tdy+ax+by+1dz&tdz

_DGtensor,M,cov_bas,cov_bas,,1,1,1,2,2,1,3,3,ax+by+1

(2.14)
M > 

VD_x,D_y,D_z

_DGvector,M,,1,1,_DGvector,M,,2,1,_DGvector,M,,3,1

(2.15)

 

We compute recurrent vector fields with respect to g3. We use the keyword argument parameters .

M > 

RecurrentTensorsg3,V,parameters=a,b,output=all

_DGvector,M,,1,1,2,_C2,3,_C1,_DGvector,M,,1,1,_DGvector,M,,1,1,2,ab,_DGvector,M,,1,1,3,1ax1,_DGvector,M,,1,1,3,1ax1,_DGvector,M,,1,1,2,ba,3,ax+by+1a2+b2ax+by+1a,_DGvector,M,,1,1,2,ba,3,ax+by+1a2+b2ax+by+1a,_DGvector,M,,2,1,3,_C1,_DGvector,M,,2,1,_DGvector,M,,3,1,_DGform,M,1,1,0,_DGform,M,1,1,0,_DGform,M,1,1,0,_DGform,M,1,3,12aax1,_DGform,M,1,3,12aax1,_DGform,M,1,3,12ax+by+1a2+b2ax+by+1,_DGform,M,1,3,12ax+by+1a2+b2ax+by+1,_DGform,M,1,1,0,_DGform,M,1,1,0,_DGform,M,1,1,0,a=0,b=0,a=0,b=b,a=a,b=b,a=a,b=0,a=a,b=0,a=a,b=b,a=a,b=b,a=0,b=0,a=a,b=0,a=0,b=0

(2.16)

 

Example 4.

We define a connection on a rank 2 vector bundle E over a 3-dimensional base manifold.

M > 

DGsetupx,y,z,u,v,E

frame name: E

(2.17)
E > 

CConnectionD_v&tdu&tdx+yD_v&tdv&tdz

_DGconnection,E,con_vrt,cov_vrt,cov_bas,,5,4,1,1,5,5,3,y

(2.18)

 

We calculate the recurrent 11 tensors on E. The command GenerateTensors is used to generate a basis for the 11 tensors.

E > 

TGenerateTensorsdu,dv,D_u,D_v

_DGtensor,E,cov_vrt,con_vrt,,4,4,1,_DGtensor,E,cov_vrt,con_vrt,,4,5,1,_DGtensor,E,cov_vrt,con_vrt,,5,4,1,_DGtensor,E,cov_vrt,con_vrt,,5,5,1

(2.19)

 

The most general 11 tensor on E is given by a linear combination of the elements of the list T, using coefficients which are functions of the base variables x, y ,z alone. We specify this dependency with the keyword argument coefficientvariables.

E > 

R4,α4RecurrentTensorsC,T,coefficientvariables=x,y,z

_DGtensor,E,cov_vrt,con_vrt,,4,5,1,_DGform,E,1,3,y

(2.20)

 

We explicitly check this result.

E > 

evalDGCovariantDerivativeR41,CR41&tα41

_DGtensor,E,cov_vrt,con_vrt,cov_bas,,4,4,1,0

(2.21)

See Also

DifferentialGeometry

Tensor

CovariantlyConstantTensors