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Connection - calculation of connection 1-forms

Connection[id]
allows one to calculate or define connection 1-forms

MORE INFORMATION

id - variable - connection identifier.

If a metric is defined, then the connection is calculated with respect to the metric, otherwise just the connection identifier is declared as a connection. If the identifier is a table then the table is declared as a connection (see example 3). To get the result of the calculation use iota operator.

Basic Examples (3)

Abstract connection (1)

Declare forms:

Declare vectors:

Declare coframe with 1-forms: e₁, e₂, ..e_n (see Coframe)

Declare frame vectors (see Frame):

Declaration of connection 1-forms:

Conformally flat metric of sphere -

(1)

Declare forms:

Declare vectors:

Declare constant :

Declare coframe:

Declare frame:

Basis one forms are not exact:

Declare conformally flat metric of S³ (see Metric):

Connection calculation:

Out[10]//MatrixForm=

Now connection 1-forms _{i, j} are calculated:

Declare forms:

Declare vectors:

Declare coframe:

Declare frame:

Connection definition:

Click for copyable input

Connection declaration:

Out[8]//MatrixForm=

Curvature calculation:

Out[10]//MatrixForm=

Torsion calculation:

Out[12]//MatrixForm=

Lie derivative calculation:

Frame Coframe Metric

Abstract calculations

Connection with torsion

Plane curves

Surface geometry

Atlas 2 - modern differential geometry

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