Cartan structural equations
To deduce the Cartan structural equations in a coordinated frame we are going to use the definition of the Christoffel symbols (connection coefficients) and where we always are going to use the Einstein sum convention:
and the curvature tensor
First, we define through the relation a set of scalar function which are easily to see that they actually are 1-forms. We observe that .
They satisfy skew-symmetry rule: , which arises from the covariant constancy of the metric tensor i.e.
that last equation is valid for each vector field , then .
Next we define through the relation
the scalars which are the so called connection 2-forms. That they are really 2-forms is an easy caligraphic exercise.
Now by the use of the Riemann curvature tensor above we see
In this last relation we recognize -in the first three terms- the exterior derivative of evaluated at i.e.
and in the last two terms its wedge product
all these for any two fields . Hence
which is called the second Cartan structural equation for the coordinated frame field .
More interesting things happen in an an-holonomic basis.
|Title||Cartan structural equations|
|Date of creation||2013-03-22 17:35:46|
|Last modified on||2013-03-22 17:35:46|
|Last modified by||juanman (12619)|