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
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.
withDifferentialGeometry:withTensor:
Example 1.
First create a 10-dimensional fiber bundle E→M 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.
g1≔CanonicalTensorsMetric,bas,3,1
g1≔dx1dx1+dx2dx2+dx3dx3−dx4dx4
Create a Riemannian metric on the fibers of E.
g1≔CanonicalTensorsMetric,vrt,6,0
g1≔du1du1+du2du2+du3du3+du4du4+du5du5+du6du6
Create a symplectic form on M.
ω1≔CanonicalTensorsSymplecticForm,bas
ω1≔dx1⋀dx3+dx2⋀dx4
Create a complex structure on the fibers of E.
ω2≔CanonicalTensorsComplexStructure,vrt
ω2≔−du1D_u4−du2D_u5−du3D_u6+du4D_u1+du5D_u2+du6D_u3
See Also
DifferentialGeometry
Tensor
GenerateForms
GenerateTensors
Download Help Document
