Define manifold as a whole

This notebook illustrates how to define a manifold as a whole using Atlas package. To do this we use 3-sphere as an example.

What we do?

    • 3-dimensional sphere is defined as an Atlas {N, S}=S3 of two charts - north (N) and south (S). Each chart has its own coframe 1-forms, frame vectors etc. One can use the mapping procedure to define chart changes and the pullback operator to transfer forms and tensors from one chart into another.
    Constants[c1,c2,...,ci,...,cn] c1,c2,...,ci,...,cn - constants identifiers.
    Curvature[id] id - variable - curvature identifier.

    Necessary functions.

Sphere:

First of all we load Atlas package:
In[7]:=
Click for copyable input
Constants declaration:
In[8]:=
Click for copyable input
Out[8]=
Forms declaration:
In[9]:=
Click for copyable input
Out[9]=
Vectors declaration:
In[10]:=
Click for copyable input
Out[10]=
Sphere dimension:
In[11]:=
Click for copyable input
Out[11]=

North chart of the sphere - N.

In[12]:=
Click for copyable input
Out[12]=
Coframe for the north chart:
In[13]:=
Click for copyable input
Out[13]=
Frame for the north chart:
In[14]:=
Click for copyable input
Out[14]=
Metric for the north chart:
In[15]:=
Click for copyable input
Out[15]=
Connection calculation for the north chart:
In[16]:=
Click for copyable input
Out[16]=
Curvature calculation for the north chart:
In[17]:=
Click for copyable input
Out[17]=
Riemanninan tensor calculation:
In[18]:=
Click for copyable input
Out[18]=
Ricci tensor calculation:
In[19]:=
Click for copyable input
Out[19]=
Ricci scalar calculation:
In[20]:=
Click for copyable input
Out[20]=

South chart of the sphere - S.

In[21]:=
Click for copyable input
Out[21]=
Coframe declaration for the south chart:
In[22]:=
Click for copyable input
Out[22]=
Frame declaration for the south chart:
In[23]:=
Click for copyable input
Out[23]=
Metric declaration for the south chart:
In[24]:=
Click for copyable input
Out[24]=
Connection calculation for the south chart:
In[25]:=
Click for copyable input
Out[25]=
Curvature calculation for the south chart:
In[26]:=
Click for copyable input
Out[26]=
Riemannian tensor calculation for the south chart:
In[27]:=
Click for copyable input
Out[27]=
Ricci tensor calculation for the south chart:
In[28]:=
Click for copyable input
Out[28]=
Ricci scalar calculation for the south chart:
In[29]:=
Click for copyable input
Out[29]=

Chart changings - and

Chart changing from S to N: .
In[30]:=
Click for copyable input
Out[30]=
In[31]:=
Click for copyable input
Out[31]=
Visualize the mapping
In[32]:=
Click for copyable input
Out[32]=
The mapping info
In[33]:=
Click for copyable input
Out[33]//TableForm=
Chart changing from N to S: .
In[34]:=
Click for copyable input
Out[34]=
Visualize the mapping
In[35]:=
Click for copyable input
Out[35]=
The mapping info
In[36]:=
Click for copyable input
Out[36]//TableForm=
Verify calculation of Riemannian and Ricci tensor using restriction operator `&/`:
In[37]:=
Click for copyable input
Out[37]=
In[38]:=
Click for copyable input
Out[38]=
In[39]:=
Click for copyable input
Out[39]=
In[40]:=
Click for copyable input
Out[40]=
For metric tensors we have:
In[41]:=
Click for copyable input
Out[41]=
In[42]:=
Click for copyable input
Out[42]=
Let us see who is who:
In[43]:=
Click for copyable input
Out[43]//TableForm=