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Tensor[CanonicalTensors] - create various standard tensors

Calling Sequences

      CanonicalTensors(keyword, spatial_type, signature, frameName)

Parameters

   keyword       - a keyword string, one of "Metric", "SymplecticForm", "ComplexStructure"

   spatial_type  - a string, either "bas" or "vrt", the spatial type of the tensor to be created

   signature     - required for the keyword "Metric", a pair of integers p (number of + 1), q (number of - 1) specifying the signature of the metric

   frameName     - (optional) a name or a string, the name of the manifold on which the tensor is to be defined

 

Description

Examples

Description

• 

This command will create the standard (flat) metric, symplectic form or complex structure.

• 

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

Examples

withDifferentialGeometry:withTensor:

 

Example 1.

First create a 10-dimensional fiber bundle EM over a 4 dimensional manifold M.

DGsetupx1,x2,x3,x4,u1,u2,u3,u4,u5,u6,M:

 

Create a metric on the tangent space of M with signature 3, 1.

M > 

g1CanonicalTensorsMetric,bas,3,1

g1dx1dx1+dx2dx2+dx3dx3dx4dx4

(2.1)

 

Create a Riemannian metric on the fibers of E.

M > 

g1CanonicalTensorsMetric,vrt,6,0

g1du1du1+du2du2+du3du3+du4du4+du5du5+du6du6

(2.2)

 

Create a symplectic form on M.

M > 

ω1CanonicalTensorsSymplecticForm,bas

ω1dx1dx3+dx2dx4

(2.3)

 

Create a complex structure on the fibers of E.

M > 

ω2CanonicalTensorsComplexStructure,vrt

ω2du1D_u4du2D_u5du3D_u6+du4D_u1+du5D_u2+du6D_u3

(2.4)

See Also

DifferentialGeometry

Tensor

GenerateForms

GenerateTensors

 


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