 CanonicalTensors - Maple Help

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

 • 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

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{Tensor}\right):$

Example 1.

First create a 10-dimensional fiber bundle $E\to M$ over a 4 dimensional manifold $M$.

 > $\mathrm{DGsetup}\left(\left[\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4}\right],\left[\mathrm{u1},\mathrm{u2},\mathrm{u3},\mathrm{u4},\mathrm{u5},\mathrm{u6}\right],M\right):$

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

 M > $\mathrm{g1}≔\mathrm{CanonicalTensors}\left("Metric","bas",3,1\right)$
 ${\mathrm{g1}}{≔}{\mathrm{dx1}}{}{\mathrm{dx1}}{+}{\mathrm{dx2}}{}{\mathrm{dx2}}{+}{\mathrm{dx3}}{}{\mathrm{dx3}}{-}{\mathrm{dx4}}{}{\mathrm{dx4}}$ (2.1)

Create a Riemannian metric on the fibers of $E$.

 M > $\mathrm{g1}≔\mathrm{CanonicalTensors}\left("Metric","vrt",6,0\right)$
 ${\mathrm{g1}}{≔}{\mathrm{du1}}{}{\mathrm{du1}}{+}{\mathrm{du2}}{}{\mathrm{du2}}{+}{\mathrm{du3}}{}{\mathrm{du3}}{+}{\mathrm{du4}}{}{\mathrm{du4}}{+}{\mathrm{du5}}{}{\mathrm{du5}}{+}{\mathrm{du6}}{}{\mathrm{du6}}$ (2.2)

Create a symplectic form on $M$.

 M > $\mathrm{ω1}≔\mathrm{CanonicalTensors}\left("SymplecticForm","bas"\right)$
 ${\mathrm{ω1}}{≔}{\mathrm{dx1}}{}{\bigwedge }{}{\mathrm{dx3}}{+}{\mathrm{dx2}}{}{\bigwedge }{}{\mathrm{dx4}}$ (2.3)

Create a complex structure on the fibers of $E$.

 M > $\mathrm{ω2}≔\mathrm{CanonicalTensors}\left("ComplexStructure","vrt"\right)$
 ${\mathrm{ω2}}{≔}{-}{\mathrm{du1}}{}{\mathrm{D_u4}}{-}{\mathrm{du2}}{}{\mathrm{D_u5}}{-}{\mathrm{du3}}{}{\mathrm{D_u6}}{+}{\mathrm{du4}}{}{\mathrm{D_u1}}{+}{\mathrm{du5}}{}{\mathrm{D_u2}}{+}{\mathrm{du6}}{}{\mathrm{D_u3}}$ (2.4)