1.3. Inner vector bundles

Starting in version 0.9, xTensor` allows the definition of vector bundles other than tangent bundles. These ``inner vbundles´´ can be real or complex, which forces the definition of the complex conjugated vbundle. Complex conjugation is controlled by the function Dagger.

DefVBundle                Define a vbundle or a sum of vbundles
UndefVBundle            Undefine a vbundle
VBundleQ                Check a vbundle
$VBundles                List of defined vbundles
$SumVBundles            List of defined sum vbundles

Definition of a vbundle.

Define a complex inner vbundle with dimension 4 (this is the fiber dimension) and base manifold M3:

In[71]:=

DefVBundle[InnerC, M3, 4, {, , ℭ, , , , , ℌ}, Dagger→Complex]

** DefVBundle: Defining vbundle InnerC.

** DefVBundle: Defining conjugated vbundle InnerC†. Assuming fixed anti-isomorphism between InnerC and InnerC†

In[72]:=

? InnerC

Global`InnerC

BaseOfVBundle[InnerC]^=M3
Dagger[InnerC]^=InnerC†
DimOfVBundle[InnerC]^=4
IndicesOfVBundle[InnerC]^={{A,B,C,D,E,F,G,H},{}}
Info[InnerC]^={vbundle,}
MetricsOfVBundle[InnerC]^={}
ObjectsOf[InnerC]^={}
PrintAs[InnerC]^=InnerC
ServantsOf[InnerC]^={InnerC†}
SubvbundlesOfVBundle[InnerC]^={}
VBundleQ[InnerC]^=True

In the process of definition, the conjugated vbundle InnerC†, with corresponding conjugated indices, has been created. It is a servant of InnerC, and hence can only be undefined through undefinition of the latter. By default, all conjugated symbols are constructed by appending the dagger character †. (This character is stored in the goblal variable $DaggerCharacter, which can be changed.)

In[73]:=

? InnerC†

Global`InnerC†

BaseOfVBundle[InnerC†]^=M3
Dagger[InnerC†]^=InnerC
DimOfVBundle[InnerC†]^=4
IndicesOfVBundle[InnerC†]^={{A†,B†,C†,D†,E†,F†,G†,H†},{}}
Info[InnerC†]^={conjugated vbundle,Assuming fixed anti-isomorphism between InnerC and InnerC†}
MasterOf[InnerC†]^=InnerC
MetricsOfVBundle[InnerC†]^={}
ObjectsOf[InnerC†]^={}
PrintAs[InnerC†]^=InnerC†
SubvbundlesOfVBundle[InnerC†]^={}
VBundleQ[InnerC†]^=True

We have already defined five different vector bundles, only one of them being direct sum of others:

In[74]:=

$VBundles

Out[74]=

{TangentS2, TangentM3, TangentM5, InnerC, InnerC†}

In[75]:=

$SumVBundles

Out[75]=

{TangentM5}

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