1.2. Product manifolds

Given two manifolds, it is always possible to define another manifold as the Cartesian product of those. The tangent bundle of the product will be then constructed as the direct sum of the tangent bundles of the submanifolds.

We define a product manifold as

In[67]:=

DefManifold[M5, {M3, S2}, {μ, ν, λ, σ, η, ρ}]

** DefManifold: Defining manifold M5.

** DefVBundle: Defining direct-sum vector bundle TangentM5.

In[68]:=

? M5

Global`M5

DimOfManifold[M5]^=5
Info[M5]^={manifold,}
ManifoldQ[M5]^=True
ObjectsOf[M5]^={}
PrintAs[M5]^=M5
ServantsOf[M5]^={TangentM5}
M5/:SubmanifoldQ[M5,M3]=True
M5/:SubmanifoldQ[M5,S2]=True
SubmanifoldsOfManifold[M5]^={M3,S2}
TangentBundleOfManifold[M5]^=TangentM5

whose tangent bundle is a direct sum of the corresponding subvbundles:

In[69]:=

? TangentM5

Global`TangentM5

BaseOfVBundle[TangentM5]^=M5
Dagger[TangentM5]^=TangentM5
DimOfVBundle[TangentM5]^=5
IndicesOfVBundle[TangentM5]^={{μ,ν,λ,σ,η,ρ},{}}
Info[TangentM5]^={vbundle,}
MasterOf[TangentM5]^=M5
MetricsOfVBundle[TangentM5]^={}
ObjectsOf[TangentM5]^={}
PrintAs[TangentM5]^=TangentM5
TangentM5/:SubvbundleQ[TangentM5,TangentM3]=True
TangentM5/:SubvbundleQ[TangentM5,TangentS2]=True
SubvbundlesOfVBundle[TangentM5]^={TangentM3,TangentS2}
VBundleQ[TangentM5]^=True

The list of product manifolds is given by another global variable:

In[70]:=

$ProductManifolds

Out[70]=

{M5}

Currently this is the only possible way of defining submanifolds of a manifold. That is, there is no code which allows you to define a submanifold of an already existing manifold. This is because that would pose the question of what is the complementary vector bundle.

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