Projective Structural Equations

A Cartan frame on a space of dimension n is presumed to be embedded in a projective space of dimension n+1. The projective frame of dimension n+1 is presumed to have a non-zero determinant everywhere (except at the origin). Hence over the punctured manifold the system is parallelizable. The Cartan structural equations vanish on n+1. However, the Cartan structural equations for the n manifold can be constructed by matrix partitions.

The idea is similar to the projective concept of defining a n dimensional subspace by means of restricting a certain function on the n+1 dimensional subspace to zero. Then the differential structure is found by computing n tangent vectors of n+1 components, orthogonal to the n+1 dimensional gradient of the constraint function. The basis frame in n+1 dimensions can then be completed by adding the (matrix) adjoint field to the n tangent vectors.


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Last update 01/23/2009
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