DefVBundle.nb_1.html

Further Examples

Load the packages:

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Before defining a vbundle we need to define a manifold:

Define a 3d manifold M:

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Its tangent bundle has been defined:

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Global`TangentM

Dagger[TangentM]^=TangentM

VBundleQ[TangentM]^=True

BaseOfVBundle[TangentM]^=M

MetricsOfVBundle[TangentM]^={}

SubvbundlesOfVBundle[TangentM]^={}

IndicesOfVBundle[TangentM]^={{a,b,c,d,e},{}}

DimOfVBundle[TangentM]^=3

Info[TangentM]^={vbundle,}

PrintAs[TangentM]^=TangentM

MasterOf[TangentM]^=M

A tangent bundle cannot be undefined, unless we undefine its base manifold:

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Now we can define inner vbundles (for example those used in gauge theories):

Define an inner vbundle over the same base manifold:

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Global`Fiber

Dagger[Fiber]^=Fiber

VBundleQ[Fiber]^=True

BaseOfVBundle[Fiber]^=M

MetricsOfVBundle[Fiber]^={}

SubvbundlesOfVBundle[Fiber]^={}

IndicesOfVBundle[Fiber]^={{A,B,C,D,F},{}}

DimOfVBundle[Fiber]^=4

Info[Fiber]^={vbundle,}

PrintAs[Fiber]^=Fiber

HostsOf[Fiber]^={M}

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We can also define a complex vbundle. Its conjugate is automatically defined. Note that the latter has its own set of abstract indices:

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Global`complex†

Dagger[complex†]^=complex

VBundleQ[complex†]^=True

BaseOfVBundle[complex†]^=M

MetricsOfVBundle[complex†]^={}

SubvbundlesOfVBundle[complex†]^={}

IndicesOfVBundle[complex†]^={{a†,b†,c†,d†,e†},{}}

DimOfVBundle[complex†]^=2

Info[complex†]^={conjugated vbundle,Assuming fixed anti-isomorphism between complex and complex†}

PrintAs[complex†]^=complex†

MasterOf[complex†]^=complex

HostsOf[complex†]^={M}

Tidy up:

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Created by Mathematica (May 16, 2008)