Atlas Paclet Symbol

Curvature - calculation of curvature 2-forms

Curvature[id]
allows one to calculate curvature 2-forms.

MORE INFORMATION

id - variable - curvature identifier.

If a connection is calculated or defined, then the curvature can be calculated completely, otherwise just the curvature identifier is declared as a curvature. To get the result of the calculation use iota operator.

The definition is as follows: , where are connection 1-forms, n is the dimension.

EXAMPLES

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Basic Examples (2)

Example 1. Conformally flat metric of sphere -

(1)

Declare forms:

Declare vectors:

Declare constant :

Declare the coframe:

Declare the frame:

Declare conformally flat metric of S³ (see Metric):

In[7]:=

Out[7]=

Connection calculation:

In[8]:=

Out[8]=

In[9]:=

Out[9]//MatrixForm=

In[10]:=

Out[10]=

Curvature calculation:

In[11]:=

Out[11]=

Now curvature 2-forms _{i, j} are calculated:

In[12]:=

Out[12]//MatrixForm=

Example 2 (1)

Declare forms:

Declare vectors:

Declare coframe:

Declare frame:

Connection definition:

Connection declaration:

In[10]:=

Out[10]=

In[11]:=

Out[11]//MatrixForm=

Curvature calculation:

In[12]:=

Out[12]=

In[13]:=

Out[13]//MatrixForm=

In[14]:=

Out[14]=

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Curvature - calculation of curvature 2-forms

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