calculate the Cotton tensor for a metric - Maple Programming Help

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Tensor[CottonTensor] - calculate the Cotton tensor for a metric

Calling Sequences

     CottonTensor(g, C, R)

Parameters

   g        - a metric tensor on the tangent bundle of a manifold

   C        - (optional) the Christoffel connection for the metric g

   R        - (optional) the curvature tensor of the metric g
   keyword  -  (optional) the keyword argument indextype = ["cov_bas", "cov_bas", "cov_bas"] or indextype = ["con_bas", "con_bas"]

 

 

Description

Examples

See Also

Description

Let Pab be the Schouten tensor for the metric gab with covariant derivative a.  The Cotton tensor is defined (in any dimension) by

Cabc=bPcacPba .

In 3-dimensions an alternative form of the Cotton tensor is a symmetric tensor density of weight one:

Yij= 12eibcgjaCabc .

 Here eiab denotes the contravariant permutation symbol, which is a tensor density of weight one.  If desired one can convert Yih to a tensor by multiplication with a suitable MetricDensity.

• 

 Cabc is completely trace-free and anti-symmetric on its last two indices; it is divergence-free on its first index. The tensor Yij is symmetric, trace-free, and divergence-free. Cabc is conformally invariant. Yij is a relative conformal invariant.

• 

If the optional arguments are not supplied, the Christoffel symbol and curvature tensor are computed directly from the metric, otherwise the supplied Christoffel symbol and curvature tensor are used.

• 

The default output is Cabc as shown above. This output can also be obtained with keyword argument indextype = ["cov_bas", "cov_bas", "cov_bas"]. The keyword argument indextype = ["con_bas", "con_bas"] returns the tensor Yij described above.

• 

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

Examples

withDifferentialGeometry:withTensor:

 

Example 1.

First create a 3 dimensional manifold M and define a metric g1 on the tangent space of M.

 

DGsetupx,y,z,M

frame name: M

(2.1)

g1 ≔ evalDGdx &t dx+xzdx &t dy+dy &t dx+y2dy &t dz+dz &t dy

g1:=dxdx+xzdxdy+xzdydx+y2dydz+y2dzdy

(2.2)

CotTen1 ≔ CottonTensorg1

CotTen1:=2y2+x2dxdxdyy5+2y2+x2dxdydxy5+x2zy+5dydxdyy4x2zy+5dydydxy42y2+x2dydydzy3+2y2+x2dydzdyy3

(2.3)

 

Check that the Cotton tensor CotTen1 is trace-free.

h1 ≔ InverseMetricg1

h1:=D_xD_xxzD_xD_zy2+D_yD_zy2xzD_zD_xy2+D_zD_yy2+x2z2D_zD_zy4

(2.4)

ContractIndicesh1,CotTen1,1,1,2,2

0dx

(2.5)

 

Check that the Cotton tensor is divergence-free on its first index.

Ch1 ≔ Christoffelg1

Ch1:=12x2zD_xdxdyy2+12x2zD_xdydxy22xzD_xdydyy+12xD_xdydz+12xD_xdzdy12xD_ydxdyy212xD_ydydxy2+2D_ydydyy+zD_zdxdxy212x3z2D_zdxdyy4+12xD_zdxdzy212x3z2D_zdydxy4+2x2z2D_zdydyy312x2zD_zdydzy2+12xD_zdzdxy212x2zD_zdzdyy2

(2.6)

ContractIndicesh1,CovariantDerivativeCotTen1,Ch1,1,1,2,4

0dxdx

(2.7)

 

Check that the Cotton tensor is a conformal invariant of the metric. We use the optional calling sequence in which the connection and curvature are specified.

g2 ≔ fx,z &mult g1

g2:=fx,zdxdx+fx,zxzdxdy+fx,zxzdydx+fx,zy2dydz+fx,zy2dzdy

(2.8)

Ch2 ≔ Christoffelg2:

R2 ≔ CurvatureTensorCh2:

CotTen2 ≔ CottonTensorg2,Ch2,R2

CotTen2:=2y2+x2dxdxdyy5+2y2+x2dxdydxy5+x2zy+5dydxdyy4x2zy+5dydydxy42y2+x2dydydzy3+2y2+x2dydzdyy3

(2.9)

evalDGCotTen2CotTen1

0

(2.10)

 

Example 2.

We continue with the manifold and metric g1 from Example 1. We check that the alternative form of the Cotton tensor is the dual of the default form of the tensor.

 

M > 

Y ≔ CottonTensorg1,indextype=con_bas,con_bas

Y:=2y2+x2D_xD_zy5+2y2+x2D_zD_xy5xx2z+5yD_zD_zy7

(2.11)
M > 

CotUp ≔ RaiseLowerIndicesh1,CotTen1,1

CotUp:=2y2+x2D_xdxdyy5+2y2+x2D_xdydxy5+xx2z+5yD_zdxdyy7xx2z+5yD_zdydxy72y2+x2D_zdydzy5+2y2+x2D_zdzdyy5

(2.12)
M > 

eps ≔ PermutationSymbolcon_bas

eps:=D_xD_yD_zD_xD_zD_yD_yD_xD_z+D_yD_zD_x+D_zD_xD_yD_zD_yD_x

(2.13)
M > 

Z ≔ ContractIndiceseps,1CotUp2,2,2,3,3

Z:=2y2+x2D_xD_zy5+2y2+x2D_zD_xy5xx2z+5yD_zD_zy7

(2.14)
M > 

evalDGYZ

0

(2.15)

See Also

DifferentialGeometry, Tensor, Christoffel, CovariantDerivative, CurvatureTensor, ParallelTransportEquations, ProjectiveCurvature, WeylTensor


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